INTRODUCTION TOMULTIDIMENSIONAL

INTEGRABLE EQUATIONSThe Inverse Spectral Transform

in 2+1 Dimensions

INTRODUCTION TOMULTIDIMENSIONAL

INTEGRABLE EQUATIONSThe Inverse Spectral Transform

in 2+ 1 Dimensions

B. G. KonopelchenkoInstitute of NuclearPhysics

Novosibirsk, Russia

Technical Editor

C. RogersLoughborougb University of Technology

Leicestershire, England

Springer Science+Business Media, LLC

Library Df CDngress CatalogIng-In-PublIcation Data

Konopelchenko. B. G. (Borls GeorgievichlIntroduction to ~ultidimensional Integrable equations : the

inverse spectral transform in 2.1 dimensions / B.G. Konopelchenko .p. c~.

Includes bibliographical references and index .

1. Inverse scattering transform. 2. Integral equations--Nu~erjcal

solutions . 3. Mathe~atlcal physiCS. I. Title.OC207 .S3K65 1992530 . , . 4--dc20 92-35653

CIP

ISBN 978-1-4899-1172-8 ISBN 978-1-4899-1170-4 (eBook)DOI 10.1007/978-1-4899-1170-4

© Springer Science+Business Media New York 1992Originally published by Plenum Press,New York in 1992.Sortcover reprint of the hardcover Ist edition 1992

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming,

recording, or otherwise, without written permission from the Publisher

To my parents

Preface

The soliton represents one of the most important of nonlinear phenomena in modern

physics. It constitutes an essentially localized entity with a set of remarkable properties.Solitons are found in various areas of physics from gravitation and field theory, plasma

physics, and nonlinear optics to solid state physics and hydrodynamics. Nonlinearequations which describe soliton phenomena are ubiquitous . Solitons and the equationswhich commonly describe them are also of great mathematical interest. Thus, the dis­covery in 1967 and subsequent development of the inverse scattering transform methodthat provides the mathematical structure underlying soliton theory constitutes one of

the most important developments in modern theoretical physics. The inverse scatteringtransform method is now established as a very powerful tool in the investigation of

nonlinear partial differential equations.

The inverse scattering transform method, since its discovery some two decades ago,

has been applied to a great variety of nonlinear equations which arise in diverse fieldsof physics. These include ordinary differential equations, partial differential equations,integrodifferential, and differential-difference equations . The inverse scattering trans­

form method has allowed the investigation of these equations in a manner comparableto that of the Fourier method for linear equations.

The majority of integrable equations treated heretofore are nonlinear differentialequations in two independent variables. In the context of evolution equations, these

usually correspond to one temporal and one spatial variable. Methods of solution ofsuch (1+1)-dimensional integrable equations are well established. These methods havebeen described in a variety of reviews and monographs. Our aim here, by contrast,

is to present the principal ideas, methods, and results concerning multidimensional

integrable equations. The generalization of the inverse spectral transform method to

nonlinear differential equations with three or more independent variables has proved to

be a far from easy task. The first such integrable equation, namely the Kadomtsev­

Petviashvili equation, was found as far back as 1974. However, an understanding of themultidimensional inverse spectral transform method has only emerged during the last

five years. The ideas proposed in that period have allowed the development of a theoryof nonlinear integrable equations in 2+1 dimensions (two spatial and one temporaldimension).

vii

viii Preface

In this monograph, both methods for the construction of multidimensional inte­

grable equations together with techniques for the calculation of exact solutions are

discussed. The a-method and nonlocal Riemann-Hilbert problem method, which cur­

rently seem the best-suited to the problem of integration of nonlinear equations in

2+ 1 dimensions, are both treated along with other approaches. It is noted that our

exposition has partly the character of a review which incorporates a wide panorama

of methods. The volume is devoted mainly to the mathematical aspects of higher­

dimensional solitons and the multidimensional inverse spectral transform method. A

discussion of the numerous physical phenomena related to multidimensional solitons

is beyond the scope of the present work. Accordingly, it is addressed, in the main,

to those who are interested in the mathematical methods of current multidimensional

soliton theory. It is intended primarily for readers who already are acquainted with the

basic elements of soliton theory. However, while such knowledge is desirable it is not

indispensable since an attempt has been made to make the exposition self-contained.

I am very grateful to V.E. Zakharov and S.V. Manakov for numerous useful dis­

cussions on multidimensional soliton theory. I am obliged also to V.G. Dubrovsky and

B.T. Matkarimov for their help in the preparation of the manuscript for print .

I express my deep gratitude to C. Rogers, who has helped to make its publicationa reality.

It is a pleasure to express my gratitude to Helen Sherwood, who typed the manu­script .

B.G. KonopelchenkoNovosibirsk

Contents

Chapter 1. Introduction

1.1. The inverse spectral transform method in 1+1 dimensions.Brief history and examples of integrable equations

1.2. Methods of solution for (l+l)-dimensional integrable equations

1.3. Multidimensional generalizations . . . . . . . . . . . . .

1

1

10

21

1.4. Methods of solution for (2+1)-dimensional integrable systems. Summary 35

Chapter 2. The inverse spectral transform method in 2+1 dimensions 47

2.1. The Kadomtsev-Petviashvili - I equation 47

2.2. The Kadomtsev-Petviashvili - II equation. Generalized analytic functions 59

2.3. Exact solutions of the Kadomtsev-Petviashvili equation 67

2.4. The Davey-Stewartson - I equation 76

2.5. The Davey-Stewartson - II equation 86

2.6. The Veselov-Novikov (NVN-I+) equation 91

2.7. The NVN-L and NVN-Io equations 101

2.8. The Nizhnik (NVN-II) equation 106

Chapter 3. Other integrable equations and methods ofsolution in 2+1 dimensions .

3.1. The multidimensional resonantly-interacting three-wave model

3.2. The Ishimori equation. The Hirota method

3.3. The Manakov-Zakharov-Mikhailov equation

113

113

116

121

3.4. Nonlocal, cylindrical, and other generalizations of the

Kadomtsev-Petviashvili equation . . . . . . . . . . . . . . . . . 130

ix

x

3.5. The Mel'nikov system .

3.6. The modified Kadomtsev-Petviashvili and Gardner equations.

The Miura transformation and gauge invariance

3.7. Further integrable equations in 2+1 dimensions

Contents

. .. 134

140

144

Chapter 4. General methods for the construction of (2+1)­

dimensional integrable equations. or-function and

8-dressing methods . . . . . . . . . . . . . . .. . . . 155

4.1. The r-function, vertex operator, and infinite-dimensional

groups for the KP hierarchy . . . . 156

4.2. Generalization of the dressing method 167

4.3. The general a-dressing method . . . 172

4.4. The a-dressing method with variable normalization 184

4.5. Operator representation of the multidimensional integrable equations 192

Chapter 5. Multidimensional integrable systems 203

5.1. The self-dual Yang-Mills equation 203

5.2. The supersymmetric Yang-Mills equation 213

5.3. Multidimensional integrable generalizations of the wave, sine-Gordon,

and self-dual equations . . . . . . . . . . . . . . . . . . . . . 218

5.4. Obstacles to multidimensionalization of the inverse spectral transform

method. I. The Born approximation 226

5.5. Obstacles to multidimensionalization of the inverse spectral transform

method. II. Nonlinear characterization of the inverse scattering data . 232

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Index 291

Chapter 1Introduction

1.1. The inverse spectral transform method in 1+1 dimensions.

Brief history and examples of integrable equations.

The first description of the soliton as a physical phenomenon was given by J. Scott­

Russell in 1843 [1]. Much later, in 1895, Korteweg and -deVries [2] derived the nonlinear

equation which describes the propagation of long water waves in a canal and which

admits the soliton solution described by Russell. This is the celebrated Korteweg-de

Vries (KdV) equationau a3u auat + ax3 + 6u ax = O. (1.1.1)

Thus, the background for the quantitative description of solitons was laid nearly a cen­

tury ago. However, the modern history of solitons is still quite recent. In 1965, Zabusky

and Kruskal showed by computer simulation that solitons of the KdV equation (1.1.1)

emerge following interaction without change of shape. Indeed their speed is likewise

unaltered [3] . An attempt to understand these unexpected and astonishing experimental

facts led, two years later, to the discovery of the inverse scattering transform methodfor the analysis of nonlinear equations such as the KdV equation which possess solitonic

behavior.

In 1967, Gardner, Greene, Kruskal, and Miura [4] demonstrated that the solution

of an initial-value problem for the KdV equation (1.1.1) is closely connected with the

one-dimensional stationary Schrodinger equation

(1.1.2)

They showed that if the potentialu(x, t) in (1.1.2) evolves in time t according to the

KdV equation (1.1.1) then the spectrum of the problem (1.1.2) is time-independent, so

that a>'jat = 0 and

(1.1.3)

1

2 Chapter 1

It turns out that equations (1.1.2) and (1.1.3) are basic to the whole procedure pro­

posed by Gardner et al. [4]. Indeed, it follows from equation (1.1.3) that the time

evolution of the scattering data for the SchrOdinger equation (1.1.2) is given by a

linear equation. Let the potential u(x, t) decrease as Ixl --+ 00 sufficiently fast and

consider the solution sp of the problem (1.1.2) such that ep --+ exp(-iAx) as x --+ -00.

As x --+ +00 one has ip --+ a(>', t)e-iAX+ b(>', t)ei>.x , where a(>', t) and b(>', t) arex .....+oo

complex functions. Equation (1.1.3) as x --+ +00, on use of this asymptotic behavior,

yields 8a/Ot = 0, 8b/Ot = 8i>.3b. Hence, the evolution of the reflection coefficient

R(A,t) := b(A, t)/a(A, t) is given by the linear equation 8R/Ot = 8i>.3R. The latter

equation is readily integrated to get R(>', t) = e8iA3R(A,0). Similarly, the evolution of

the scattering data which corresponds to the discrete spectrum of (1.1.2) is again given

by a linear equation.

The preceding observations indicate the following procedure for solution of the

Cauchy problem for the KdV equation:

I II { 8iA3t } IIIu(x, 0) --+ {R(A, 0), .. .} --+ R(>', t) = e R(A,0), . .. --+ u(x, t). (1.1.4)

At Stage I, given u(x, 0), one must calculate the inverse data {R( >., 0),>'n, bn(O)} at

t = 0, that is, one must solve the forward scattering problem for (1.1.2). Stage II is

straightforward, on use of the properties of the scattering data mentioned above. In

Stage III, given the scattering data, one must reconstruct the potential u(x, t) , that is,

one must solve the inverse scattering problem for the stationary Schrodlnger equation(1.1.2). The solution of the inverse scattering problem for equation (1.1.2) is given by

the well-known Gelfand-Levitan-Marchenko linear integral equations

du(x, t) = 2dx K(x, x, t),

K(x,x',t)+F(x+x',t) +100

dsK(x,s)F(s+x',t) =0,

x' > x

(1.1.5a)

(1.1.5b)

(1.1.5c)

Accordingly, the solution of the Cauchy problem for the nonlinear KdV equation is

reduced to a sequence of linear problems. Stage III in (1.1.4) is the pivotal one, being

the most technical and .complicated. Indeed, it gave the original appellation, namely

the inverse scattering transform (1ST) method to the whole procedure indicated in

(1.1.4).

In general, the solutions of the forward and inverse scattering problems for (1.1.2)

cannot be obtained explicitly. However, the application of the 1ST method allows \1', to

Introduction 3

treat the initial value problem for the KdV equation with some measure of completeness.

In particular, this method provides explicit expressions for a remarkable class of exact

solutions to the KdV equation, namely, the multisolitons. These correspond to the

case R == 0, when F(x, t) adopts the form F(x, t) = E~=1 bn(0)e8>'~t->'nX. In this

instance, equation (1.1.5b) is a Fredholm integral equation with degenerate kernel and

is readily solved to yield u(x, t) = 2d2/dx2 ln det A(x), where A(x, t) is the Nx N matrix

with elements Anm = onm+ [bn(O)bm(O)/(>'n +>'m)]e8>'~t-(>'n+>'m)X. In the simplest case

N = 1, one has the famous KdV soliton: u(x, t) = 2>'~ cosh-2{..\1(x-4>'~t -xo)}, where

Xo = [1/2>'d In[b1(0)/2>'1] ' It should be emphasized that these solitons are essentially

nonlinear objects.

An operator formulation of the ideas of Gardner et al. in [4] was subsequently given

by Lax in [5]. Therein, it was shown that the KdV equation (1.1.1) is equivalent to the

operator equation8Lat = [L,A], (1.1.6)

where L = -82/8x2 -u(x, t) and A = -483/8x3 -3 (u8/8x + 8/8xu) -4i>.3. Equation

(1.1.6) is now known as the Lax equation, and the pair of operators L and A is usually

referred to as a Lax pair. This formulation manifestly reflects the fact that the evolution

of the potential u(x,t) according to the KdV equation (1.1.1) represents an isospectral

deformation of the operator L. Lax also showed that, on appropriate choice of the

operator A, one can represent in the form (1.1.6) a broader class of nonlinear differential

equations associated with the spectral problem (1.1.2), namely, the so-called KdV

hierarchy. It is noted that the operator equation (1.1.6) represents the compatibility

condition for the system of two linear equat ions (L - ).,2)'l/J = 0 and (Ot + A)'l/J = 0 and

is equivalent to the system (1.1.2), (1.1.3).

A second important nonlinear differential equation integrable by the 1ST method

was found by Zakharov and Shabat [6]. Thus, in 1974, they showed that the one­

dimensional nonlinear Schrodinger (NLS) equation

(1.1.7)

is also representable in the Lax form (1.1.6). The operator L in this case is

L = i (~1 ~) :x+(~ ~q), so that, instead of (1.1.2), one has the one-dimensional

stationary Dirac equation

8'l/J = i>. (1 0) 1/J + i (0 q(x, t) ) 1/J8x 0 -1 ij(x,t) 0 '

(1.1.8)

4 Chapter 1

(1.1.9)

where 1/J = (1/J1, 1/J2)T and the bar indicates complex conjugation. Zakharov and Shabat

investigated the forward and inverse problems for the spectral problem (1.1.8) . They

also calculated the multisoliton solutions of the NLS equation (1.1.7) and studied their

collision properties. An important feature of the technique as introduced by Zakharov

and Shabat in [6] is that the equations of the inverse scattering problem they derived

constitute a system of singular integral equations.

The work of Zakharov and Shabat stimulated an intensive search for other nonlinear

equations integrable by the 1ST method . In 1972, Wadati [7] had applied the 1ST

method to the modified KdV (mKdV) equation

8u 83u 28uat + 8x3 + 6u 8x = o.

The corresponding spectral problem in this case is of the form

81/J = i).. ( 1 0) 1/J + (0 u) 1/J.8x 0 -1 u 0

In 1973, Zakharov and Shabat [8] treated the NLS equation (1.1. 7) with a negative sign

attached to the nonlinear term and nonvanishing asymptotic values of q(x, t) . Zakharov

[9] subsequently demonstrated the applicability of the 1ST method to the Boussinesq

equation1 3 2

Utt - Uxx - 4UxXXX + 4(U )xx = 0, (1.1.10)

where Ut := 8u/at, Ux := 8u/8x. The corresponding spectral problem in this case isthe third-order differential spectral problem

( 83

( 3) 8 3 3) 3- + 1 - -u - - -Ux + -w 1/J = ).. 1/J8~ 2 8x 4 2

where w is an appropriately chosen function of u.

The two-component generalization of the NLS equation, namely,

was investigated by Manakov [10] . At about the same time, Zakharov and Manakov [11]

showed that systems of equations which describe the resonant interaction of three-wave

packets, such as8q1 8q1 __7ft + V1 8x + 1'1 q2q3 = 0,

8q2 8q2 __7ft + V2 8x + 1'2q1qs = 0,

8q3 8q3 __at + V3 8x + 1'3q2q1 = 0,

(1.1.11)

Introduction 5

are also integrable by the 1ST method . In this case, the spectral problem adopts the

following 3 x 3 matrix form:

(1.1.12)

where f3ik = (~ - ak)' and al > a2 > a3. In the same year, 1973, Ablowitz, Kaup,Newell, and Segur (AKNS) [12] showed that the 1ST method is also applicable to the

sine-Gordon equation

CPEr, + sin cP = 0, (1.1.13)

where cp(e, "I) is a scalar function and e,"I are cone variables. Equation (1.1.13) is

equivalent to the compatibility condition for the linear system

81/1 . ( 1 0) i (0 CPE )8e = ZA 1/1 + '2 1/1,.. 0 -1 CPE 0

(1.1.14)

81/1 i (cos cP, - i sin cP )8"1 = 4A i sin cP, _ cos cP t/J.

Subsequently, the sine-Gordon equation in the form

CPtt - CP:z;:z; + sin sp = 0 (1.1.15)

was subjected to the 1ST method [13,14]. As with equation (1.1.13), the auxiliary

linear problems for (1.1.15) are of 2 x 2 matrix type, but the dependence on thespectral parameter A is more complicated. During this period, various properties of

these integrable equations were investigated (see, for example, [15-23]) . The review

[23] excited particular interest in soliton problems.

It has since become clear that the 1ST procedure is applicable to a wide class of

nonlinear equations. Importantly, these include model equations not only of intrinsic

physical importance but also of great universality. For instance, the KdV equation

(1.1.1), originally derived in the context of nondissipative water waves [2], has been

subsequently used in the study of a number of phenomena with both nonlinearity and

dispersion, such as ion acoustic waves in plasma, waves in anharmonic lattices, hydro­

magnetic waves, waves in mixtures of liquid and gas bubbles, longitudinal dispersive

waves in elastic rods, and self-trapping of heat pulses in solids (see [23)). The NLS

equation (1.1.17), on the other hand, has been applied to the study of self-focusing of

optic beams, one-dimensional self-modulation of monochromatic waves, and Langmuir

waves in plasma [23]. The sine-Gordon equation has appeared in the study of the

6 Chapter 1

propagation of flux in Josephson tunnel junctions, plastic deformation of crystals, as

well as the propagation of Bloch walls in magnetic crystals [23].

Since 1974, the number of papers devoted to the study of the structure and prop­

erties of integrable equations has increased manyfold. Two papers of those published

in 1974 have had a profound influence on the subsequent development of the inverse

scattering transform method. These were the fundamental papers of Zakharov and

Shabat [24] and Ablowitz, Kaup, Newell, and Segur [25]. In [241, the first general

approach to the simultaneous construction of integrable equations and their solutions

was formulated. The starting point of the procedure involves the factorization of an

integral operator on the line as the product of two Volterra type integral operators.

These Volterra operators are then used to construct dressed differential operators L

starting with an initial operator Lo. This dressing method allows one to construct pairs

of commuting differential operators Li ;L2 with

(1.1.16)

and to calculate the solutions of the system of nonlinear equations equivalent to the

operator equation (1.1.16). The general forms of the operators Ll and L2 discussed in

[24] are as follows:N

L1 = a8y + L un(x,y,t)8;,

n=O

M

£2 = flat + L vm(x,y, t)a;;,m=O

where 8x := 8/8x, 8y := 8/8y and un(x, y, t), vm(x,y, t) are matrix-valued functions of

the independent variables x, y, t while a, /3 are constants. The commutativity operator

representation (1.1.16) of the integrable equations for such operators L) and L2 is

equivalent to the Lax representation (1.1.6).

Another approach was proposed in the celebrated AKNS paper [25]. Therein

the problem of delineating those nonlinear equations which arise as the compatibility

condition of the linear system

881/J = i>. (1 0) 1/J + (0 q(x, t) ) 1/J,x 0 -1 r(x,t) 0

81/J = (A(X,t,>.), B(X,t,>.») 1/J8t C(x, t, >'), D(x, t, >')

(1.1.17a)

(1.1.17b)

was considered. In the above, q(x, t), rex, t) are scalar complex functions while A, B, C,

D are functions of x and t with polynomial dependence on the spectral parameter >.. The

Introduction 7

problem (1.1.17a) represents a straight-forward generalization of the problem (1.1.8).

The equality of the cross derivatives in x and t, that is, the compatibility condition for

the system (1.1.17), allows one to express A,B,C, and D via q(x,t) and r(x,t) and,

thereby, to arrive at a system of two equations for the two functions q and r [25]. In

particular, in the case of A, B, C, and D, which are quadratic in A, the system

iqt + qxx + 2q2r = 0,(1.1.18)

is obtained. For r = ±ij this system reduces to the NLS equation (1.1.7).

For functions A, B, C, D cubic in A, one obtains the system

(1.1.19)rt + ru x + 6qrrx = O.

In the particular cases r = 1 and r = q, the system (1.1.19) produces , in turn, the KdV

and mKdV equations. In the case r = a + {3q where a and (3 are constants, the system

(1.1.19) is equivalent to the Gardner equation qt + qxxx + 6aqqx + 6{3q2 qx = O. The

latter represents a combined KdV and mKdV equation.

The paper [25] also initiated the development of an important procedure for the

description of hierarchies of the integrable nonlinear equations, namely the so-called

recursion operator method.

A feature of the class of integrable equations considered above, with the exception

of the sine-Gordon equation, is that they do not contain the independent variables x and

t on an equal footing. This characteristic is also inherent in the approaches proposedin papers [241 and [25].

The next important steps in the development of the 1ST method are to be foundin the papers [26, 27] . Firstly, Zakharov and Shabat extended the class of the auxiliary

linear problems in the 1ST method. They considered linear systems of the form

at/;ax = u(x, tj A)t/;,

at/;at = v(x, t j A)t/;,

(1.1.20a)

(1.1.20b)

(1.1.21)

where u(x, tj A) and v(x, t; A) are N x N matrix-valued functions of x and t with rational

dependence on the spectral parameter A. The compatibility condition for the system

(1.1.20) is of the formau av- --+[u v]=Oat ax ' ,

that is, the commutativity condition [L1 ' L2 ] = 0 for the operators £1 = ax - u(x, t; >')

and L2 = at-v(x, t) . Equation (1.1.21) is equivalent to a system of nonlinear differential

8 Chapter 1

equations for the coefficients of the expansions of u and v. A basic feature of equations

(1.1.21) and the auxiliary problems (1.1.20) is that they now contain the independent

variables x and t on an equal footing. The relativistically-invariant two-dimensional

principal chiral field model equation

(1.1.22)

where g(x, t) is a non degenerate N x N matrix, is the simplest important example of

such a type of equation. Equation (1.1.22) is equivalent to the compatibility condition

of the linear system [27]8'I/J = _g-lgx'I/J8x ).+ 1 '

8'I/J = g-lgt 'I/J.at ).-1

(1.1.23)

The pivotal idea in paper [26] was the introduction into the 1ST method of a

new mathematical construction which involves the Riemann-Hilbert conjugation prob­

lem. This idea was to have a crucial influence on the subsequent development of

the 1ST method. The Riemann-Hilbert problem consists of the following. Given the

function G()'), defined on the contour r , which divides the whole complex plane into

two regions D+ and D-, one seeks to construct an analytic function 'I/J().) such that

its boundary values 'I/J+()') and 'I/J-().) on the contour r are related by the equality

'I/J+()') = 'I/J-().) + 'I/J-()')G()'). A solution of the matrix Riemann-Hilbert problem isgiven by linear singular integral equations. The use of the Riemann-Hilbert problem

gives rise to a very simple and effective method for the calculation of the exact multi­

soliton solutions for equations representable in the form (1.1.21) [26, 27].

A great number of nonlinear differential equations in 1 + 1 dimensions have been

investigated by the different versions of the 1ST method. In addition to the integrable

equations named above, we shall find occasion in the sequel to discuss several further

important examples.

It is noted that equation (1.1.22) contains interesting two-dimensional relativistic­

invariant systems as special reductions [27]. In particular, it includes the n-fieldequation

(1.1.24)

where nn = 1, and the Dodd-Bullough-Zhiber-Shabat-Mikhailov equation

(1.1.25)

The applicability of the 1ST method to equations (1.1.24) and (1.1.25) was first demon­

strated in [28] and [29]. The reduction problem, that is, the problem of transition in

Introduction 9

the 1ST method from generic integrable systems to integrable systems with a smaller

number of the dependent variables is an important one (see, for example, [30, 31]).

Among the many integrable equations of considerable physical interest one must

also cite both the Maxwell-Bloch system

E" -p=O,

1 -Ne + "2 (pE + pE) = 0,

pe-NE=O,

(1.1.26)

which describes the propagation of light pulses in two-level media [18, 32, 33], and the

one-dimensional continuous isotropic ferromagnet Heisenberg model equation

8s ~ ~ 08t + 8 X 8 x x = , (1.1.27)

(1.1.28)

where ss = 1. Equation (1.1.27) is equivalent to the compatibility condition for the

system [24, 35]

~~ = ~ (sa)1/J ,

81/J 0,2(_).,. A(~ ~)(_) .,.at = -2 8(1 'fI - 2" 8 x (1 8(1 'fI'

where (1t, (12, and (13 are the usual Pauli matrices . In fact , equation (1.1.27) and the

NLS equation (1.1.7) are gauge equivalent [36]. This notion of gauge equivalence, which

was introduced simultaneously with the Riemann-Hilbert problem, is an important

ingredient of the 1ST method [26, 27, 36].

The one-dimensional Landau-Lifshitz equation

8s ~ ~ ~ y~ 08t + 8 X 8 x x + 8 X 8 = , (1.1.29)

where Y is a constant 3 x 3 diagonal matrix, describes the anisotropic continuous

ferromagnet and is also integrable by the 1ST method [37,38]. However, in this case the

2 x 2 matrices u and v in the auxiliary problems (1.1.20) are not rational, but rather, are

elliptic functions in the spectral parameter A. This leads to the necessity of considering

the Riemann-Hilbert problem on a torus [39, 40].

A generalization of the scheme (1.1.20) is required if the equation

(1.1.30)

where G(~, TJ) is a 2 x 2 nondegenerate matrix and g := det G, is imbedded in the 1ST

method. Equation (1.1.30) is equivalent to an Einstein equation wherein the metric

tensor depends only on two independent variables [41, 42]. It is representable as the

10 Chapter 1

commutativity condition [L1' L2] = 0 of two operators L1,L2 which contain, in addition

to the derivatives ae and aT/' derivatives in the spectral parameter A [41]. Equivalently,

equation (1.1.30) is representable in the form (1.1.21) with A a certain function of ~ and

TJ [42]. The development of these ideas has led to the generalization of the 1ST method

to the case of a variable spectral parameter [43].

Overall, the theory of nonlinear integrable equations with two independent variables

now incorporates a broad variety of different methods and approaches. It includes

generalizations of the original inverse spectral transform method on the whole line to

the periodic case [44-47] and more specialized procedures, such as the Hirota formalism

[19, 48, 49], the resolvent method [5D-52], the direct linearization method [53-55], the

r-function method [56, 57], together with various algebraic and algebraic-geometry

techniques [58-77] .

Many surveys and monographs have been devoted to nonlinear integrable equations

in 1 + 1 dimensions. Methods for their solution, the structure and properties of these

equations and also their detailed history are described in [23, 78-122]. These texts

are recommended to those interested in the underlying theory and literature on one­

dimensional solitons.

Here, we shall confine ourselves to those basic elements of the theory of the (1+ 1)­

dimensional integrable equations, their methods of construction and solution that bear

upon the multidimensional generalizations to be considered in this monograph.

1.2. Methods of solution for (l+l)-dimensional integrable equations

The starting point for the application of the 1ST method to (1 + I)-dimensional

nonlinear equations is their representation as the compatibility condition for the system

of two linear equationsL11/J = 0,

(1.2.1)

The commutativity condition

(1.2.2)

(1.2.3a)

is the typical operator form of the compatibility condition for the system (1.2.1) and,

as such, is the usual operator form of integrable equations in 1 + 1 dimensions.

The most common operators L1and L2 which are met in (1+1)-dimensional soliton

theory are either differential operators of the form

N

L 1 = -A + L un(x, t)a;,n=O

M

L2 = at + L vm(x, t)a; ,m=O

(1.2.3b)

Introduction

where Un and Vrn are matrix-valued functions, or the so-called rational operators

11

(1.2.4)

where Unt(x, t), Vmt(x, t), uS (>" t), vS (>" t) are matrix-valued functions. The pairs of

operators (1.2.3) and (1.2.4) are not entirely unrelated. Thus, each pair can be obtained

from the other by an appropriate reduction or limiting process.

The two versions of the dressing method, considered in [24] and [26], correspond to

these two types of operators £1 and £2. In both cases, the infinite-dimensional analogs

of the well-known Gaussian decomposition of a matrix into the prod uct of two triangular

matrices is used for the construction of the operators £1 and £2 and of the solutions

of the corresponding integrable equations. In the first case (1.2.3), this reduces to the

problem of the decomposition of an integral operator on a line into the product of two

Volterra-type operators [241 . In the case (1.2.4), it reduces to the Riemann-Hilbert

problem for the representation of a function given on a contour in the complex plane

into the product of two functions which are analytic on the different sides of the contour

(26).

Here, we summarize the basic elements of the dressing method in these two forms.

Let us start with the case (1.2.3) [24] . Let F be the integral operator on the line:

1+ 00

(F"p)(x) = -00 dzF(x, z)"p(z), -00 < x < 00. The analog of the Gaussian problem

corresponds to representations of the operator F in the form

(1.2.5)

where 1<+ and 1<- are the Volterra-type integral operators:

1± 00

(I<±"p)(x) = ±:r dzK±(x, z)"p(z).

Let us multiply (1.2.5) by 1 + 1<+ from the left and put first z > x and then z < x.

Thus, one obtains, in turn,

F(x, z) + K+(x, z) +100

dsK+(x, s)F(s, z) = 0,

z>x

K-(x, z) = F(x , z) +100

dsK+(x , s)F( s, z ).

z < x

(1.2.6a)

(1.2.6b)

12 Chapter 1

The integral equation (1.2.6a) and the formula (1.2.6b) give the solution of the factori­

sation problem (1.2.5).

Next, let Lo be the differential operator which commutes with the operator

F : [F, Lo] = O. With the use of (1.2.5), this condition implies

(1.2.7)

In the generic case, the operator, which is obtained from the differential operator by

a similarity transformation with the Volterra-type operator, contains the integral part.

However, the equality (1.2.7) guarantees that the operator L defined by (1.2.7) is a

purely differential operator. Condition (1.2.7) gives also the relations between theN

coefficients of the operator L = -A + L Un (x)a; and the coefficients of the initialn=O

N

operator Lo = A+ L uno(x)a;, namely,n=O

UN = UNO, UN-l = UN-l ,O

(1.2.8)

where ~k = (!... - !...)k K+(x, z)/ . Here we have restricted our attention to theax az x=z

scalar case.

The connection between these constructions and integrable nonlinear differentialequations in 1+ 1 dimensions is made if one introduces parametric dependence on timet into all the above quantities and then considers the pair of differential operators LlO

N

and L20 in the form LlO = -A + L uno(x, t)a~ and L20 = at + Ao which commuten=O

with the operator F:[LlO , FI = 0, [at + Ao, F] = 0

and also commute with each other so that

[LlO, at + Ao]= o.

(1.2.9)

By virtue of (1.2.7), the dressed operators L; and L2 = at + A also commute, whence

This guarantees the compatibility of the system

(at + A)t/J = 0,

(1.2.10)

(1.2.11)

Introduction 13

where t/J = (1 + k-)t/Jo and the function t/Jo obeys the compatible system of equations

Ltot/Jo = 0,

(8t + Ao)t/Jo = o.

Condition (1.2.10) yields a system of nonlinear evolution equations for the coefficients

un(x, t) in the operator Ll. Solutions of this nonlinear system may be constructed by

the above dressing procedure. First, one finds the kernel F(x, z) of the operator Ffrom the equations (1.2.9). Then equations (1.2.6) allow, in principle, the calculation of

the kernels k±(x,z). Finally, using the dressing formulae (1.2.8) (which are especially

simple in the case when all UnO == constant), one finds the Un (x, t). The functions

un(x, t) calculated in this way obey equation (1.2.10) by construction. Thus, the dress­

ing method incorporates simultaneously a procedure for the construction of integrable

equations and a method for calculation of solutions.

In the particular case L IO = -8; - >.2, L20 = 8t + 48~, the dressing method gives

the KdV equation (1.1.1) with u(x, t) = 2d: K+(x, x, t), and equation (1.2.6a) reduces

to equation (1.1.5b).

Note that in this version of the dressing method, the role of the spatial variable x

which appears in the factorization problem (1.2.5) is essentially distinguished from the

role of the time variable t which is a parameter in the problem (1.2.5).

An entirely different situation arises in the second version of the dressing method

connected with (1.2.4). In this case, the basic ingredient of the method, namely the

Riemann-Hilbert problem, is formulated in an auxiliary space, namely, on the plane of

complex variable >., and the variables z and t appear in this problem as the parameters.

The matrix local Riemann-Hilbert problem is formulated as follows [26, 94, 123,

124]. Let a simple closed contour r be given on the complex A-plane . r divides the

entire plane into two parts D+ and D- (0 E D+, 00 E D-), and the N x N matrix

function G(>') without singularities is given on this contour r. The problem is to find

the function X(>') analytic everywhere outside r such that the equality

(1.2.12)

holds on the contour r where X+ and X- are the boundary values of the function X(>')

on r for the regions D+ and D-, respectively. In such a formulation, the solution of

the Riemann-Hilbert problem is obviously non-unique since, together with the solution

X(>.), the function g(>.)X(>'), where g(>.) is an arbitrary nondegenerate N x N matrix,

is also a solution. In order to remove this non-uniqueness, it is necessary to normalize

the Riemann-Hilbert problem by fixing the value of the function X(>') at one point. The

usual canonical normalization is X(>' = 00) = 1. Note also that (1.2.12) also represents

a factorization problem since 1 + G(>') = (X-)-I(>.)X+(>').

14 Chapter 1

(1.2.13)

The Riemann-Hilbert problem is said to be regular if det X(A) :f. 0 everywhere in

D+ U D-. If det X(A) = 0 at a finite number of points in D+ U D-, then the problem

(1.2.12) is referred to as a singular Riemann-Hilbert problem or a Riemann-Hilbert

problem with zeros.

The solution of the Riemann-Hilbert problem may be reduced to the solution of a

system of linear singular integral equations on the contour r [26, 94, 123, 124]. Here,

for simplicity, we restrict ourselves to the regular Riemann-Hilbert problem with the

canonical normalization. It follows from well-knowntheorems in complex variable theory

that, outside the contour r, the function X(A) may be represented in the form

1 f , k(A')X(A) = 1+ 27l"i Jr dA A' - A'

where k(A) = X+(A)-X-(A), A E r . On the contour r the use of the Sokhotsky-Plemelj

formulae gives± 1 l ' k(A')

X (A) = 1+ -2' dA AI A '0 '7l"Z r - =FZ

Substitution of (1.2.14) into (1.2.12), yields

{I f , k(A') }

k(A) = 1 + 27l"i Jr

dA A' _ A+ iO G(A),

(1.2.14)

(1.2.15)

The integral equation (1.2.14) and formula (1.2.13) (or the integral equation (1.2.15)

and formula (1.2.13)) give the solution of the regular Riemann-Hilbert problem. The

solvability conditions for the regular matrix Riemann-Hilbert problem are quite compli­

cated (see for example [123, 124]). However, if the Riemann-Hilbert problem is solvable

then it is uniquely solvable (in the fixed normalization) .

For the Riemann-Hilbert problems with zeros, one must define, in addition, the set

of discrete variables. The solution of this problem is also given by a system of linear

singular integral equations [26, 94, 123, 124].

As in the first version of the dressing method, the Riemann-Hilbert problem pro­

cedure allows one to conjugate solutions of the system of equations representable in the

form (1.2.2) with the operators £1 and £2 of the type (1.2.4).

Let t/JO(A) be some solution of the system (1.2.10), that is, (1.1.20) where uo and

Vo are solutions of equation (1.2.2), that is (1.1.21). Let r be an arbitrary contour with

GO(A) a matrix function on r. We define the function

G(x, t ;A) = t/Jo(x, t, A)GO(A)t/JOl(X, t, A) (1.2.16)

for all x and t. Now let us consider the Riemann-Hilbert problem (1.2.12) with the

function G(x, t; A) given by (1.2.16). The solution X of this problem will also depend

on the variables x and t. It may happen that the contour r passes through the poles

Introduction 15

of the functions Uo and vo. At these points, we set G == 1. Acting on (1.2.12) with a",. 01/10 .1. bt .since ax = UO'f"O, we 0 am

A E f. (1.2.17)

It follows from (1.2.17), that the matrix u(x, tj A) admits continuation from the contour

r to the whole complex plane and u(x, tj A) is the rational function whose poles coincide

with the poles of the function uo(x, t; A). Similarly, acting on (1.2.12) with at gives

(1.2.18)

The poles of the function v(x, t; A) defined by (1.2.18) coincide with the poles of the

function vo(x, t; A). It is readily seen that the functions u(x , t j A) and v(x, t;A) con­

structed in this way together with the function 1/1 = X+,¢o (or the function 1/1 = X-'¢o)

obey the linear system1/1", = u,¢,

(1.2.19)'¢t = v'¢.

Since the functions u and v have poles at the same points as Uo and Vo, the coefficients

in the expansion of u and v (1.2.4) obey the same system of equations as the coefficients

of Uo and Vo .

Thus, the formulae (1.2.17) and (1.2.18) are nothing but the dressing formulae

analogous to those given in (1.2.8). For a concrete dressing procedure, one must know

the starting functions Uo, Vo, '¢o and the solution of equation (1.2.15). Note that

equations (1.2.19) and (1.1.21) have the trivial solution Uo = A(x, A), vo = B(t, A),where [A,B] = o. This solution is very convenient as a starting point.

It is emphasized that the above dressing procedure is purely local with respect to

the variables x and t and does not impose any restriction on the behavior of the functions

u and v as Ixl -+ 00. Choosing the contour I' and the function G one can construct

a wide class of exact solutions, such as solutions u, v which exponentially increase at

Ixl-+ 00.

In order to have a complete description of the class of solutions which tend asymp­

totically to Uo and Vo at [z] -+ 00, one must consider the Riemann-Hilbert problems

with zeros. The most interesting case corresponds to G == O. In this case, the solution

of the regular Riemann-Hilbert problem is trivial, the functions X+ and X- are rational

functions and the whole dressing procedure becomes purely algebraic. Solutions of the

integrable equations constructed with the use of such a special Riemann-Hilbert problem

are usually referred to as the soliton solutions. Applications of the Riemann-Hilbert

problem to the calculation of exact solutions of integrable equations is considered in

detail in the monographs [94, 111].

16 Chapter 1

At the present time, the method based on the use of the Riemann-Hilbert problem

is the most potent and comprehensive method of solution of initial value problems for

integrable nonlinear differential equations in 1+1 dimensions [94, 97, 111]. To illustrate

this, following [94] we consider the KdV equation and reformulate the inverse scattering

problem for the stationary Schrodinger equation (1.1.2) as a Riemann-Hilbert problem.

First, in addition to the solution cp(x, A) of the problem (1.1.2) such that

cp --+ exp( -iAX) we introduce the solution 'l/J given by its asymptotic behavior at%-+-00

x - +00 : 'l/J --+ exp(-iAx) . Since the solutions 'l/J and ij; form a complete basis (for%-+-00

real u),

cp(X,A) = a(A)'l/J(x, A) + b(A)ij;(X, A). (1.2.20)

The functions a(A) and b(A) are the elements of the scattering matrix. It happens that

it is much more convenient to deal with the solutions of (1.1.2) normalized to unity at

infinity. For this reason, we introduce the functions x+ = cpeiAX and X- = 'l/Jei AX• These

obey the equation

8;X± + u(x)x±+ 2iA8xX± = 0, (1.2.21)

where 8x == :X.In terms of X±, the equality (1.2.20) may be re-written as follows:

ImA = 0, (1.2.22)

where R(A) := ~~~~ . For real u(x), from the definition of X- and equation (1.2.21),

one has X(x,A) = X(x, -A) at ImA = O.

A simple way to ascertain the analytic properties of the functions X+ and X- is

to use the linear integral equations obeyed by these functions. Taking into account the

fact that X± --+ 1, it may be verified that equations (1.2.21) are equivalent to the%-+-00

following integral equations:

where the Green functions G± for the operator 8; + 2iA8x are

G±(x A) = O(±x)(1 _ e2iAX)'2iA .

(1.2.23)

(1.2.24)

Here and below, {}(~) is the Heaviside function: {}(~) = {~: ~ ~ ~. Hence, the integral

equation (1.2.23) is a Volterra type equation. This is an important feature of the one­dimensional case.

For the Green function G+, the exponent decreases for ImA > 0, and for G-,

it decreases for ImA < O. As a consequence, the function X+ and cp admit analytic

Introduction 17

continuation into the upper half-plane 1m>. > 0 and the function X- and 'I/J admit

analytic continuation into the lower half-plane 1m>' < o. From equation (1.2.23) it

follows, in particular, that

.1+00

a(>.) = 1 - 2Z>.

-00 dxu(x)x+(x, >.),

and

or

lim x-(x, >') = 1 - 2i, 100

dx'u(x')A.....OO 1\ X

() 2. 1. (,8X-(X,>.))u x = - Z 1m 1\ 8 .A"'"00 X

(1.2.25a)

(1.2.25b)

It is known from the theory of the stationary Schrodinger equation (1.1.2) that

the function a(>.) may have a finite number of simple zeros which are situated on the

imaginary axis in the upper half-plane. Hence, X+(x, >')ja(>.) is an analytic function in

the upper half-plane with a finite number of simple poles.

Thus, we arrive at the problem of the construction of the function X(x, >') which

is analytic in the upper half-plane and lower half-plane, has a finite number of poles,and which has a jump across the real axis equation to R(>.)e2iAXx-(x, ->'). This is the

standard singular Riemann-Hilbert problem. The contour r is the real axis and the

normalization is the canonical one. The solution of this Riemann-Hilbert problem is

given by a formula of the type (1.2.13), namely,

N 2iAn X - ( ') 1 1+00 R( ") - ( ") 2iA' x( ') = 1 ""' cne X x, -I\n _ d" 1\ X x, -1\ e (1 226)

XX,I\ +~ '_' +2 . 1\ >.'_>. , ..n=1 1\ I\n rrz -00

where Cne2iAnXx-(x, ->'n) is the residue of the function X(x, >') at the point >'n . The

formula (1.2.26) in the limits Imx -+ -0 and>. -+ ->'m produces the system of equations

[941

N 2iAn X - ( >')-( >') = 1 + ""' cne X x, - n

X x, L...J >. _ >'nn=1

1 1+00 ,R(>")e2iA' x x-(x, ->.')

+ 2rri -00 d>' >.' - >. + iO '

N 2iAn X _( >')-( ->. ) = 1 - ""' cne X x, - n

X x, m ~ >'m + >'n

Imx = 0 (1.2.27)

m = 1,2, . . . , N (1.2.28)

18

On use of (1.2.25) and (1.2.27), one obtains

Chapter 1

The system of equations (1.2.27)-(1.2.29) comprises a closed set of linear singular equa­

tions which gives the complete solution of the inverse scattering problem associated with

the spectral problem (1.1.2). Given the scattering data {R(A), (ImA = 0, An,en}, in

principle, the potential u(x) may be reconstructed.

It has been seen that, for the KdV equation, the scattering data evolve linearly

in time t. The formulae (1.2.27)-(1.2 .29) give the solution of the stated initial value

problem for the KdV equation.

Pure soliton solutions correspond to the case R == O. In this case, the system

(1.2.28) becomes purely algebraic. It is easily solved and we obtain the usual multi­

soliton solutions of the KdV equation.

The preceding formulae have been presented, in part, to allow subsequent compar­

ison with their two-dimensional generalizations.

The treatment of the KdV equation via the Riemann-Hilbert problem which has

been presented here is closely connected with other approaches. Thus, it is noted that

equation (1.2.27) can be rewritten in the equivalent form

ImA = 0where

{R(-A)dA,

dp(A) =27ricn6(A - An)dA, ReA = O.

(1.2.30)

Let us introduce the function k(x, s) via the formula X-(x , A) = 1+idS k(x, s)e- iA(s-x).

Substituting this expression into (1.2.30) and taking the integral -.!..j+oodAe-iA(X-X1) ,

27r -00

we obtain the Gelfand-Levitan-Marchenko equation (1.1.5b) where F(x) =

2~ ( dp(A)ei Ax and u(x) is given by the formula (1.1.5a). Hence, the original versioni:of the inverse scattering transform method for the KdV equation [4, 21] is equivalent to

that considered above.

The integral equation (1.2.30) admits natural generalization to the case of an

arbitrary contour r and arbitrary measure dp(A). The corresponding equation (1.2.30)

is precisely the integral equation used in the direct linearization method [53-55].

In what follows, we give a short review of the main properties of (1+ 1)-dimensional

nonlinear integrable equations. The most remarkable property of such equations is

Introduction 19

the existence of classes of multisoliton-type solutions. These describe interaction of

solitions. This interaction is of an unusual nature. In the simplest cases, for instance

for the KdV, NLS, and sine-Gordon equations, the velocities and profiles of the solitons

remain unchanged and only a phase shift appears after collision. This phase shift may

be calculated explicitly [5, 6, 7, 14, 94, 97, 111]. A more interesting situation arises for

those nonlinear systems integrable via higher order spectral problems. For example, for

the three-waves system described by equations (1.1.11), decay processes 1 -+ 2 + 3 and

junction processes 2+ 3 -+ 1 become possible [11, 22, 94, 125, 126]. More complicated

processes similar to the transition of n solitons into m solitons exist for systems of

N(N -1) resonantly interacting waves [94] and for the principal chiral fields model [94,

127]. However, all these non trivial processes have a rather special structure. They are,

in fact, the processes of gluing and regluing of a number of elementary solitons. The

multisoliton solutions of the integrable equations in 1+ 1 dimensions correspond to the

discrete spectrum and they may be calculated explicitly within the framework of the

1ST method [94, 95, 97-101, 111].

A special feature of the dynamics of integrable equations is closely connected with

the existence of infinite sets of conservation laws a~n + a:Cn= 0 (n = 1,2, . ..) and,

1+ 00

correspondingly, of integrals of motion Cn = -00 dxTn (x, t) for these equations. Higher

conservation laws for the KdV equation have been calculated in the paper [15]. The

1ST method provides a standard method for the calculation of all integrals of motion

via the recurrence relations which result from the spectral problems [94, 95, 97-101].

A voluminous literature exists on the problem of the existence of higher integrals of

motion, to their structures and properties, to restrictions on the dynamics which are the

consequence of the existence of the higher integrals of motion and to the classificationof the nonlinear equations which possess such integrals of motion (see, for example,

[128-145]).

Integrable equations also possess infinite-dimensional symmetry groups. These

symmetry transformations contain derivatives and nonlinearities of higher orders and arenot the usual contact transformations. A study of such transformations (Lie-Backlund

transformations) has led new developments in the group-theoretical analysis of differ­

ential equations (see, for example, [146-162]). Effective methods have been developed

for the classification of certain classes of (1+ 1)-dimensional nonlinear equations which

possess higher symmetries (see, for example, [163-167]) .

Besides infinite-dimensional symmetry groups, the admittance of Backlund trans­

formations is generic for the nonlinear equations integrable by the 1ST method . For

the sine-Gordon equation, such a transformation was found in the last century in

connection with the study of surfaces of constant negative curvature [168]. Similar

Backlund transformations also exist for the other integrable equations listed in the

20 Chapter 1

previous section. A remarkable feature of the Backlund transformation is that they

have corresponding nonlinear superposition formulae which allow the construction of the

multisoliton solutions of the nonlinear equations in a purely algebraic manner. Methods

of construction of Backlund transformations for different equations, their structure and

properties, and the interrelation between the Backlund transformations and the 1ST

method have been extensively discussed (see, for example, [81, 169-193]) .

Nonlinear evolution equations integrable by the 1ST method possess a further

interesting and important property. They are Hamiltonian systems. Hamiltonian

structures of the integrable equations have been studied, starting with [16, 194J, in

a number of papers (see, for example, [111, 195-203]). Indeed, Magri discovered that

some of the (1 + I)-dimensional integrable equations are two-fold (in fact infinitely­

fold) Hamiltonian systems [204J. An important notion in the Hamiltonian treatment of

integrable equations is the classical r-matrix introduced in [211J. This classical r-matrix

plays a fundamental role in the orbit-algebraic approach to the Hamiltonian formalism

(see, for example, [111, 211-217]).

Some nonlinear integrable equations are not only Hamiltonian systems but are also

completely integrable systems in the Liouville sense (see [218]) . The complete inte­

grability of the KdV equations has been proved in Zakharov and Faddeev's pioneering

paper [16J . For the NLS equation this has been established in [195], for the sine-Gordon

equation in [14J, and for the system of three resonantly interacting waves in [2I9J .

Complete integrability has been also proved for other (1 + I)-dimensional integrable

equations (see, for example, [111, 201] .

Finally, one of the most striking features of the (1 + I)-dimensional nonlinear

equations integrable by the 1ST method is that these equations and their properties

possess an explicit recursive nature. For instance, the KdV equation and all higher

KdV equations can be represented in the following compact form:

where a== :x and the operator A is

n = 1,2,3, ... (1.2.31)

(1.2.32)

The KdV equation itself corresponds to n = 1 and Wi = -4. The form (1.2.31)

of the KdV hierarchy and the operator (1.2.32) was first introduced by Lenard (see

[21]). The operator A which plays a key role in the formula (1.2.31) is the so-called

recursion operator. This recursion operator A also allows one to represent, in compact

form, the higher symmetries of the KdV equation Dn u = €n Anau (n = 1, 2,3, . . .) androo

the integrals of motion en = L oo dx(A+t+iu where A+ is the operator adjoint to A.

Introduction 21

The recursion operator A+ also occurs in the description of the hierarchy of the

Hamiltonian structures for the KdV equation [204, 205]. The generalization of the

recursion operator to the case of the two different potentials u and u' allows one to

construct and represent, in the compact form, the general Backlund transformation for

the KdV equation [220] . One of the most important properties of the recursion operator

is that the square eigenfunction ep(x, A) of the problem (1.1.2) is its eigenfunction (see,

for example, [100])

and in the absence of solitons

2' /+00u(x, t) = ~ dA AR(A,t)ep2(x, t, A).7r -00

(1.2.33)

(1.2.34)

The formulae (1.2.33) and (1.2.34) allow us to regard the mapping R(A,t) -+ u(x ,t)and the whole 1ST method as a nonlinear Fourier transform [25, 100]. In that context,

the recursion operator A+ is the analog of the operator :x (the momentum) and the

function ep2(x,t, A) plays the role of plane waves.

The work of Ablowitz, Kaup, Newell, and Segur [25], and Calogero and Degasperis

[221, 222] has played a significant role in the formulation of the theory of recursive

operators. Subsequent developments of the recursion operator method play an impor­

tant part of the theory of the (1 + I)-dimensional integrable equations. Indeed, it is a

very convenient tool for the investigation and description of the group-theoretical and

Hamiltonian structures of a wide class of the (1+ I)-dimensional integrable equations

(see, for example [100, 118, 210, 223-247]).

This concludes our summary of the basic elements of the theory of (1+ I)-dimen­sional integrable equations . We now pass to an overview of the nonlinear integrable

equations in three or more independent variables which will be our main concern in this

text.

1.3. Multidimensional generalizations

The first generalization of the 1ST method to nonlinear equations which involve

functions of three independent variables (x, y, t) was given by Zakharov and Shabat

in [24]. The dressing method proposed therein is well-adapted to the construction of

classes of (2 + I)-dimensional nonlinear integrable equations and to the generation of

solutions. Indeed, introduction into the factorization formalism (1.2.5) of the parameter

y along with the parameter t (time) and starting with commuting operators £10 and

£20 of the form

22 Chapter 1

(where £01(8x ) and £02(8x ) are, in general, matrix-valued differential operators in x),

we obtain, on repetition of the dressing procedure as described in the previous section,

the dressed differential operators L1 and L2

L1 = 8y + £1(8x ) ,

L2 = 8t + £2(8x ) ,

N

where £1 = LUn(x,y,t)8~ and £2n=O

operators L1 and L2 commute, that is,

M

= Lvm(x,y,t)8;a. It is noted that thesem=O

(1.3.1)

The latter commutativity condition is equivalent to an integrable system of nonlinear

equations in 2+ 1 dimensions (x, y, t) . Indeed, two-dimensional generalizations of a wide

class of (1 + I)-dimensional integrable equations are representable in the form (1.3.1).

The Kadomtsev-Petviashvili (KP) equation

(1.3.2)

where u(x, y, t) is a scalar real function, was the first such integrable (2+ I)-dimensional

equation to be constructed. It describes two-dimensional waves propagating in the

x direction with slow variation in the y direction [248J. The KP equation is a two­

dimensional generalization of the KdV equation (1.1.1). The operators L1 and L2 for

the KP equation are [24, 249]

L1 = 0'8y + a; + u(x, y, t),

L2 = 8t + 4~ + 6u8x + 3ux - 3a(8;1 Uy ) .

(1.3.3)

The properties of the KP equation depend critically on the sign of 0'2 . For the KP-I

equation (0' = i) the problem L 11/J = 0 is the nonstationary one-dimensional Schrodinger

equation while for the KP-II equation (0' = 1) it is the heat equation. The KP equation is

probably the most studied integrable equation in 2+ 1 dimensions. This is, in particular,

because it provides a good testing ground for the two-dimensional version of the 1ST

method.

A number of important (2+ 1)-dimensional integrable equations are associated with

various choices of the scalar differential operators L1 and L2. In particular, the modified

KP equation [57, 520]

(1.3.4)

Introduction 23

has £1 = 8y +8i - 4u8x • It is noted that the two-dimensional integrable generalization

of the Boussinesq equation coincides with the KP equation with the interchange y +-+ t

[24].

The two-dimensional integrable analog of the system (1.1.18) is the following system

[251,252]

where

irt - Dl r - rsp = 0,

D2r,p + 2Dl (rq) = 0,

a2

Dl = "4 {a2(b- a)8; + 2a(a + b)8x8y + (b - a)8;} ,

D2 = (a8x + 8y)(-a8x + 8y) = 8; - a28;,

a2 = ±1,

(1.3.5)

a and b are arbitrary constants.

The system (1.3.5) is equivalent to the commutativity condition for the operators

[251,252]

£1 = 8y + a (~

£2 = 8t - i (~

o ) o, + (0 q) ,-1 r 0

0) 2 .b-a (0 q) (AI'8x + t-2- 8x +

bar 0 A3 ,

(1.3.6)

where All A 2 , A 3 , At are appropriate functions of q and r , In the particular case

a = -b = 1, r = ij, the system (1.3.5) reduces to

iqt + ~(qxx + a2qyy) + Iql2q+ q¢ = 0,

¢yy - a2¢xx - (a2 - 1)lq!;x+ (a2+ 1)lql~y = 0,

(1.3.7)

where ¢ = r,p -lqI2 . For a = i, equation (1.3.7) is the Davey-Stewartson (DS-I) equation

which describes two-dimensional long surface waves on water of finite depth [253].

Equation (1.3.7) is a two-dimensional integrable generalization of the NLS equation

(1.1.7).

Another interesting case of the system (1.3.5) is the specialization [252]

iqt + q~~ + qr,p = 0,(1.3.8)

24 Chapter 1

which arises in the degenerate case b = 0, a = -2 with (72 = 1, r = q following the

change of independent variables given by o~ = Oy + Ox, oTt = Oy - Ox. Equation (1.3.8)

arises in plasma physics.

The 1ST method is also applicable to the equations which describe resonant three­

wave interaction in multidimensional space, namely, [254]

oqt + -" + - - 07f; V1 v q1 1'1 q2q3 = ,

Oq2 + -" + - - 07ft V2 vq2 1'2qtq3 = , (1.3.9)

OQ3 - V - - 07ft + V3 Q3+ 1'3Q1Q2 = ,

where V is the gradient operator and ih, fh, V3 are arbitrary multidimensional vectors.

In fact, the multidimensional nature of the system (1.3.9) is illusory. The system (1.3.9)

really belongs to a system with only three independent variables. Thus, in terms of the

characteristic coordinates Xi (i = 1,2,3) defined as usual by Oi == 00 := 00 + ViV, theXi t

system (1.3.9) is of the form

(1.3.10)

(1.3.12)

03Q3 + 1'3q1 q2 = 0

with corresponding auxiliary linear system [56, 255]

Ok'l/Ji + 1'kQj'l/Jk = 0,(1.3.11)

Oi'l/Jk + 1'iQj'l/Ji = 0,

where the indices i, k,j are cyclic and run through the values 1,2,3. The use of

characteristic coordinates reduces the calculations and allows detailed investigation of

both the forward and inverse scattering problem for (1.3.11) [255, 256].

Among other (2+ I)-dimensional integrable systems which possess the commutative

operator representation (1.3.1), one should also note the two-dimensional integrable

generalization of the isotropic Heisenberg model (1.1.27) . This extension is described

by the system [257]

- - (- 2-) - - 0St + S x Sxx + (7 Syy + CPySx + CPxSy = ,

CPxx - (72cpyy + 2(72s.(Fx X S'y) = 0,

where ss = 1 and (72 = ±1. The corresponding operators £1 and £2 are of the form

[257]

(1.3.13)

Introduction 25

where P(x, y, t) = as(x, y, t) . The equations (1.3.12) describe the classical spin system in

the plane. This system is of interest for at least two reasons. Firstly, t he system (1.3.12)

possesses the topological invariant Q = 4~ JJdx dy S.(sx x Sy) and its solutions may

be classified according to the values of Q (0, ±1, ±2, . ..). Secondly, the system (1.3.12)

is gauge equivalent to the DS system (1.3.5).

The two-dimensionalization of pairs of operators of the type (1.2.3) and of the type

(1.2.4) leads to equations representable in the form (1.3.1). Indeed , the first procedure

for the two-dimensionalization of such operators is achieved by the formal substitution

,X -+ 8y together with u(x, t), v(x, t) -+ u(x, y, t) , v(x , y, t) where y is a new independent

variable. As a result, we again arrive at (1.3.1) up to the change x ...... y. It is emphasized

that, in this extension, both operators L 1 and L 2 become partial differential operators.

There is also another method of two-dimensionalization. It consists in the intro­

duction of a new variable only into the second operator L2 while leaving the operator

L1 as an ordinary differential operator. This was first proposed by Calogero [258]. It

was subsequently generalized in [29] . Nontrivial nonlinear equations in 2+ 1 dimensionsM

arise in this context if one chooses L1 = L(8x ) -,x, L2 = 8t + j('x)8y + L vm(x, y, t)8:;m=O

M

or, equivalently, L2 = 8t + j(L)8y + L vm(x,u, t)8:;, where j(L) is some function andm=O

L(8x ) is an ordinary differential operator.

Two simple examples of integrable equations representable in the form [L1 ' L2] = °with operators L1 and L2 of this type have been found by Zakharov. The first example

is the two-dimensional generalization of the KdV equation in the form [2601

(1.3.14)

for whichL1 = 8; + u(x, y, t) - ,x,

(1.3.15)

where j = 6au + 4,8(8;lU y ) , 9 = 30ux + 3,8uy , and 0,,8 are arbitrary constants.

The second equation is [252]

iu, + ux y + uw = 0,

with (1.3.16)

26

For the latter system the operators L1 and L2 are

L 1 = i (1 0) ax + (0_ u) _x,o -1 -u 0

{(1 0) (0 u) } 1(iw,L2 = at - 2 i ax + _ ay + 2 _o -1 -u 0 2uy,

Chapter 1

(1.3.17)

The system (1.3.17) represents, like (1.3.7) and (1.3.8), a two-dimensional integrable

generalization of the NLS equation (1.1.17).

The above second procedure for two-dimensionalization admits a more general

formulation [26, 260]. Thus, one can consider a pair of commuting operators of the

form3

L1 = L ai(>.)aX i + u(x, >.),i=1

(1.3.18)3

L2 = L bk(>.)aX k + vex, >.),k=1

where u(x, >') and vex, >') are matrix-valued functions and ai(>'), bk(>') are commuting

matrices with polynomial dependence on the spectral parameter >..

A simple and elegant (2+ 1)-dimensional integrable equation with operators of the

form (1.3.18) is the equation [30, 261]

(1.3.19)

where g(x,y,t) is a nondegenerate N x N matrix, ~ = ~(x+(1Y), Tf = ~(x - (1y), and

(12 = ±1. The corresponding operators L 1 and L 2 are (for the case (1 = i, see [261])

L1 = >.2aT/ - a~ + >.g-1gt + g-1g~,

L2 = >.aT/ +at +g-1gt.(1.3.20)

The two methods of two-dimensionalization outlined above (see [252, 260]) repre­

sent ways of introducing a new independent variable which preserve basic features of

the standard theory of (1 + I)-dimensional integrable equations. Thus, they preserve

the operator representation of the integrable equation as the compatibility condition

for a system of two linear equations L1'l/J = 0, L2'l/J = 0, namely [L1, L2] = O. In fact,

the commutativity requirement [L1, L 2] = 0 is a very strong condition. It guarantees

the existence of a common spectrum for the operators L1 and L2. In the case L1 =

L - >., L2 = at+A, this condition ensures the time independence of the whole spectrum

of the operator L .

Introduction 27

The characteristic feature of all the partial differential operators Ll and L2 con­sidered heretofore is that, while these operators contain partial derivatives in all three

variables, only partial derivatives with respect to one of the independent variables is

greater than one. This is the penalty for the commutativity requirement [L1, L2] = O.

Indeed, if one tries to extend the class of operators L; and L2 to include higher order

derivatives in two or three variables, then it is readily seen that the commutativity

condition [Ll, L2] = 0 is satisfied only if the coefficients of these operators are constantsor if the operators L}, L2 are of the form (1.3.18). In broad terms, this is due to the

fact that the number of conditions which arise from the commutativity requirement isgreater than the number of the coefficient functions in the operators Ll and L2. Thus,

the condition [Ll, L2] = 0 is too restrictive and cannot produce nontrivial evolutionequations when the operators L} and ~ contain higher-order derivatives in two or

three independent variables.

In order to circumvent this problem one must relax the condition [L1, L2] = O.

In fact, it is only required that the equations L}t/J = 0, L2t/J = 0 should have asufficiently broad family of common solutions. For one-dimensional operators L] of

the form L1 = L(8x ) - A, the necessity of the condition ~~ = 0 for all A is dictatedby the fact that, in this case, one must know the scattering data for all A in order to

reconstruct the potential, via, for instance, formulae (1.1.5). This is not necessary in the

multidimensional case. Indeed, the inverse scattering problem for the multidimensionaloperator can be solved even if the spectral characteristics are collected from the solutions

which correspond to only one value Ao of the spectral parameter A (see, for example,

[262]) . Hence, in the two-dimensional case, it is sufficient that the system (L - Ao)t/J =

0, L2t/J = 0 should be compatible for a least one eigenvalue of the operator L. In

other words, the operator L2 should leave invariant the subspace of eigenfunctions t/Joof the operator L associated with the eigenvalue Ao so that the condition L}L2t/JO = 0is satisfied [263]. This condition is fulfilled if

(1.3.21)

where B is an appropriate operator. The operator equation (1.3.21) is a generalization

of the commutativity condition which guarantees the existence of a sufficiently broad

family of common solutions for the pair of equations L}t/J = 0 and L2t/J = O.

The equation (1.3.21) is the triad operator representation first introduced by Man­

akov [263] . In the case L} = L - Ao, L2 = 8t + A, equation (1.3.21) leads to ageneralization of the Lax equation (1.1.6), viz,

Lt = (L - Ao)A - (A + B)(L - AO) '

This important observation by Manakov has played a significant part in the subsequent

development of the two-dimensional version of the 1ST method. The requirement

28 Chapter 1

(1.3.21) is less restrictive than the commutativity condition and allows us to construct

nontrivial integrable nonlinear equations when the operator Ll contains second order

derivatives in two independent variables. The first example of such an equation, albeit

of complicated form, was presented in [2631.

The most important equation in 2+ 1 dimensions representable in the form (1.3.21)

is the equation

(1.3.22)

(1.3.23)

(1.3.24)

where u(x, y, t) is a scalar function, k1, k2 are arbitrary constants, o~ = Ox - aoy , 0.., =

Ox + a0'll' and a 2 = ±1. For this equation

L1 = -(0; - a20; ) + u(x,y, t),

L2 = Ot + k10l + k20~ - 3k1(oi1u..,) .0.., - 3k2(0;;IU~) .0~ ,

B = -3(kl(oi1u..,.., + k2(0;;lu~~)) .

Equation (1.3.22) was first introduced by Nizhnik [2641 in the case a = 1 and by Veselov

and Novikov [265] in the case a = i, k1= k2 = 1. It will be subsequently referred to as

the Nizhnik- Veselov-Novikov (NVN) equation. The NVN equation represents another

two-dimensional generalization of the KdV equation. In contrast to the KP equation

(1.3.2) and equation (1.3.15) this equation contains the spatial variables x and y in a

symmetric manner.

Another interesting integrable system which possesses the triad representation

(1.3.21) adopts the form

qt + !:1q - Q(q~)2 + (3(q..,)2 - 2QO;;IU~ + 2(30i1u.., = 0,

where q(e, 11, t) and u(e, 11, t) are scalar functions, Q and (3 are arbitrary constants, and

!:1 = QOl- (30~. In this case [266]

L1= o..,o~ + q~o.., + U,

L2 = Ot + Qot + (30~ + 2(3q..,0.., + 2Q(0;;lu~),

B = 2!:1q.

The system (1.3.24) contains a two-dimensional integrable generalization of a dispersive

long wave system, namely

(1.3.25)Ut + u~~ + 2(uv)~ = 0,

Introduction 29

as introduced by Boiti, Leon, and Pempinelli [267], and corresponding to the particular

case a = -1, (3 = 0 with v == qE'

A further example of a nonlinear system in 2+ 1 dimensions with the triad operator

representation is the system [268]

acpy + CPu + ucp = o.

Here u(x, y, t) is a real function while cp(x, y, t) is complex-valued, a2 = ±1 and 7" = ±1.

For this system

4cp )48x '

and the operator representation is of the form

where

Other integrable nonlinear system which possess the triad operator representation

will be presented in Chapter 3.

The operator triad representation (1.3.21) has two important properties. Firstly,

the operator equation (1.3.21) is equivalent to the system [L1, L211/1 = 0, L11/1 =

O. The use of such a weak commutativity condition, namely the commutativity on a

submanifold of solutions, allows us to construct nonlinear integrable equations in 2+ 1

dimensions [267] . Secondly, given the integrable equation, the operators L2 and B in

the triad representation (1.3.21) are defined nonuniquely, but up to the transformation

L2 -+ L~ = L2 + CL1, B -+ B' = B + [L1 ,C], where C is an arbitrary differential

operator [252]. This freedom allowsus to choose the operators L2 and B in a convenient

form. Moreover, for equations (1.3.22), (1.3.24) and other equations with associated

scalar triad representation of the type (1.3.21), there exists an equivalent commutativity

representation [LfI, L~] = 0 with appropriate matrix operators Lfl and L~ [266].

The method of generalization connected with the transition from the commutativity

representation [L1' L2] = 0 to the triad representation [L1' L2] = BL l has a natural

30 Chapter 1

extension. Thus, an algebraic form of the compatibility condition for the system L1.,p =0, L2.,p = 0 more general than (1.3.21) is the quartet operator equation

where B 1 and B2 are appropriate differential operators. Such an operator equation

appears as the compatibility condition for operators L1 and L2 of the form L, =

A.(8X3)8xi + B,(8x3) (i = 1,2) ,where A,(8x3), B,(8x3) are matrix-valued operators.

In this case, B1= D1(8X3)8x2+C1(8X3) and B2 = D2(8x3)8xl +C2(8x3), where D" C,are certain operators. Such operators L1 and L2 arise after the application of the first

method of multidimensionalization (A -+ 8X3) to the operators (1.2.4) with generic

rational dependence on A [260].

An example of a nonlinear integrable system which possesses the quartet operator

representation is given by the system

(1.3.26)

where u(x, y, t) and <p(x, y, t) are scalar functions, 8t; = 8x - (J8y, 8TJ = 8x + (J8y, and

(J2 = ±1. For this system, the operators L1 , L2, Bv, B2 are given by

It is noted that the (2 + I)-dimensional integrable equations listed above possess

an important feature. Thus, they are integrodifferential equations which involve an

arbitrary auxiliary function. The corresponding integrals, are evidently non-unique.For instance, for the KP equation, the integral 8;1 may be defined by (8;1 f)(x, y) =

1:00 dx' f(x', y)+a(y) , where a(y) is an arbitrary function, while for the NVN equation,

the integrals 8i1, 8;1 are defined up to arbitrary functions a(7J) and b(~). This

arbitrariness in the definitions of the integrals and auxiliary functions means that,

Introduction 31

in fact, we have infinite families of integrable equations parameterized via arbitrary

functions. This plurality is not spurious but rather, with different choices ofthe arbitrary

functions, gives rise to different auxiliary linear problems L2'l/J = 0 and, consequently,

to different time-evolutions of the same initial data. However, in the sequel, all such

arbitrary "integration" functions are set identically equal to zero.

The (2 + I)-dimensional equations considered in this section up to now have in

common one principal feature, namely, as for the (1 + I)-dimensional case, they are

integrable via two auxiliary linear problems L 1'l/J = 0, L 2'l/J = O. The two-dimensional

linear problems can be generated from one-dimensional spectral problems with appropri­

ate matrix structure. In particular, this has been established for the KP equation [269] .

In such cases, the linear problems L1'l/J = 0, L2'l/J = 0 and the associated integration

procedure for corresponding integrable equations in 2+ 1 dimensions inherit fundamental

properties of (1 + I)-dimensional integrable systems.

However, a quite new situation arises when one considers truly three-dimensional

problems which contain the independent variables on an equal footing. It can happen

that, in this case, it is not sufficient to consider only two compatible auxiliary linear

problems in order to obtain a nontrivial nonlinear equation. A simple example illustrates

this point. Let us consider the two auxiliary problems

(1.3.27a)

(1.3.27b)

where Ui ad Vi are matrix-valued functions. The compatibility condition for the system

(1.3.27), (namely, the condition 8X 38X ! 8X 2 'l/J = 8X 28x ! 8X 3 'l/J) gives

(1.3.28)

All derivatives 8x ! 'l/J, 8X 2 'l/J, 8X 3 'l/J, 8x 28x 3 'l/J in (1.3.28) are independent. The condition

that the coefficient 8x 28x 3 'l/J in (1.3.28) vanishes gives U2 = V3 := ¢. Similarly, the

condition that the coefficients of 8x 2 'l/J and 8x 3 'l/J vanish gives V1 = -8X 3 ¢.¢- 1 and

U1 = -8X 2 ¢.¢- 1 respectively. The latter expressions for U1 and V1 show that the

coefficient of 8x ! 'l/J is identically zero. Thus, the compatibility condition for the system

(1.3.27) does not give any nontrivial nonlinear equation.

On the other hand, the situation changes radically if one demands that the equation

(1.3.29)

holds in addition to equations (1.3.27). In this case, the derivative 8x 28x 3 'l/J is not

independent. On substitution of expression for 8x 28x 3 'l/J as given by (1.3.29) into

32 Chapter 1

equation (1.3.28), we arrive at nontrivial relations between the functions Ui , Vi, Wi.

The compatibility conditions for the three linear problems (1.3.27a), (1.3.27b), and

(1.3.29) provide the following nonlinear system

(i,k,l= 1,2,3), (1.3.30)

where U:k = - ::~g;l +gi Ai~\k«'. [Ai, Ak] = 0 (i, k = 1,2,3), Ai (i = 1,2,3)

are arbitrary constants, and gi(Xl, X2, X3) are N x N matrices. Note that there is no

summation on repeated indices in (1.3.30). The matrices U~k and Ufk correspond to the

representation of the linear problems (1.3.27), (1.3.29) in the compact form

(i,k=I,2,3j i:f:k).

The system (1.3.30) is, in fact, a three-dimensional integrable generalization of the

principal chiral fields equation (1.1.22) and was introduced in [270, 271].

The nonlinear system (1.3.30) is equivalent to the system of operator equations

3

[Li, Lk] = L 'YiklLll=l

(i,k=I,2,3), (1.3.31)

where 'Yikl are certain operators [272]. This operator system is the algebraic form of the

compatibility condition for the linear system (1.3.27) and (1.3.29). Thus, novel algebraic

structure make their appearance in the case of the truly three-dimensional integrablesystems.

It turns out that the system (1.3.30) is typical of the class of truly three-dimensional

integrable systems. In general, nonlinear integrable systems with three independent

variables typically are equivalent to the compatibility condition for three linear auxiliary

problems Li(oXl' OX2' oxa)'l/J = 0 (i = 1,2,3). The operator form of this compatibility

condition is (1.3.31). The main feature of the integration procedure for such genuine

three-dimensional integrable equations is that all three linear problems Li'l/J = 0 (i =

1,2,3) must be solved simultaneously in contrast to the (2+1)-dimensional cases which

inherit the standard properties of the integrable (1 + I)-dimensional cases wherein one

must solve the forward and inverse spectral problem only for £1'l/J = 0 while L2'l/J = 0

defines the time evolution of the corresponding inverse data.

The operator representation (1.3.31) of the integrable equations is a logical devel­

opment of the original idea of commutative operator representations due to Lax. The

generic three-dimensional case is also important because it contains integrable systems

which are not straightforward generalizations of (1+1)-dimensional integrable equations.

The structure and properties of such generic three-dimensional integrable equations are

of considerable current interest.

Introduction 33

The properties of nonlinear integrable equations in 2 + 1 dimensions are, in many

respects, similar to the properties of the (1+ I)-dimensional integrable equations. Thus ,

(2 + I)-dimensional integrable equations possess soliton-type solutions. These solitons

interact in a similar manner to their (1+1)-dimensional counterparts. Moreover, certain

of these equations (KP-I, DS-I, NVN-I) have solutions of a novel type called lumps.

These are nonsingular solutions which decay algebraically at infinity. Lumps emerge

completely unaltered following collision. Even the usual phase shift associated with

(1 + I)-dimensional soliton interaction is absent.

As in the (1+ I)-dimensional case, integrable equations in 2+1 dimensions possess

infinite sets of integrals of motion, infinite-dimensional symmetry groups, and Biicklund

transformations. These equations represent Hamiltonian systems for which, in many

cases, complete integrability has been established. Moreover, hierarchies of (2 + 1)­

dimensional integrable equations may be generated by a single recursion operator as

in the (1 + I)-dimensional case. In 2 + 1 dimensions, this recursion operator is bilocal

one. Integrable equations in 2 + 1 dimensions also admit novel features such as, for

instance, degenerate dispersion laws. Overall, the richness of the theory of (2 + 1)­

dimensional integrable equations bears comparison with that of (1 + I)-dimensional

integrable systems.

A completely different situation obtains for such multidimensional integrable equa­

tions with four and more independent variables as are known at the present time.

All these systems are equivalent to the commutativity condition [Li, Lk] = 0 (i, k =

1, . . . ,d) of operators of the form

d

L, = LAi (k) (>')8 z k +Ui(Xl, ... ,Xd j>'),

k=l

(1.3.32)

where Ai(k) are commuting matrices with polynomial dependence on the spectral param­

eter >. and Ui(Xj >') are matrix-valued functions ofthe d independent variables Xl, ... ,Xd

and x.The most important known multidimensional integrable system is the self-duality

equations [273, 274]

(IL, v = 1,2,3,4) (1.3.33)

of classical Yang-Mills theory. Here FIJIJ = 8zl'AIJ - 8zvAIJ + [AIJ ,AIJ], where AIJ is

a matrix-valued vector-potential and clJlJpr is a completely antisymmetric unit tensor.

The system (1.3.23) is equivalent to the commutativity of the two operators [273]

(1.3.34)

34 Chapter 1

where 'Vp. = ax,. + Ap. (J.L = 1,2,3,4). A considerable number of papers have been

devoted to the self-dual Yang-Mills equations (1.3.33). This is due to the fundamental

role which gauge fields play in the modern theory of elementary particles.

The self-dual equations (1.3.3) possess a number of interesting properties. In par­

ticular, equations (1.3.33) possess the celebrated instanton solutions, namely, rational

nonsingular solutions with finite action [275]. Equations (1.3.33) also admit remarkable

structure most conveniently described by methods of algebraic geometry [276] . Despite

such a richness of properties, the self-dual Yang-Mills equations, nevertheless, bear the

impress of two dimensionality. This is connected both with the special complex structure

of equations (1.3.33) and with their equivalence to the commutativity condition of the

two operators (1.3.34).

Another example of multidimensional integrable system which is equivalent to the

commutativity condition of two partial differential operators is the system [277]

which describes N2 - N resonantly interacting waves in d-dimensional space (N > d).Here Paa = 0, VaP(l) = Wa(l) - aa(l)naP and

naP = Wa(l) - Wp(l) = Wa(2) - Wp(l) = ... = Wa(N) - WP(N).

aa(l) - ap(l) aa(2) - ap(2) aa(N) - ap(N)

The operators Ll and L2 are given by

(a,j3 = 1,. . . ,N)

(1.3.36)

where AlaP = aa(l)Dap, BlaP = Wa(l)Dap and QaP = na {3 Pa {3 '

The multidimensional generalizations of the wave and sine-Gordon equations, con­

nected with imbedding in multidimensional Riemann spaces, represent multidimensional

integrable systems of another type [278]. These d-dimensional systems are equivalent

to the commutativity conditions [Li , Lk] = 0 of d first-order differential operators

L i (i = 1, . . . , d) with the matrix coefficients [278] . Finally, it is noted that one

of the supersymmetric generalizations of the Yang-Mills equations (with N = 3,4)

is representable as the commutativity condition of the three first-order differential

operators [279] .

The brevity of the list of the nonlinear integrable systems with four or more inde­

pendent variables reflects the difficulties which appear in their construction. These are

connected, in particular, with algebraic constraints which arise from the compatibility

Introduction 35

conditions as well as restrictions related to the equations which characterize the inverse

problem data in multidimensional scattering problems.

1.4. Methods of solution for (2+1)-dimensional integrable systems.

Summary

The original version of the dressing method based on the use of the factorization

method for integral operators on a line [24] is also applicable to those (2+ I)-dimensional

equations which are representable in the form (1.3.1). In this case, the factorizable

integral operator P is required to commute with both of the undressed differential

operators £10 and £20. The solution of the inverse problem is given by the integral

equations (1.2.6) together with the dressing formulae (1.2.8).

Novel solutions of the KP-I equation may be constructed within the framework of

this procedure [280] . These are the so-called lump solutions. The simplest one-lump

solution of the KP-I equation (1.3.2) adopts the form

(1.4.1)

where x = x - 3(a2 + b2)t - xo, jj = y + 6at - Yo, and a,b,xo,Yo are arbitrary real

constants . It follows from (1.4.1) that the lump has no singularities for all x,y, and

t and decreases in all directions on the plane as 0 (l/x2, l/y2) at [z], Iyl -> 00 and

moves with the velocity v= (v:r , vy), where V:r = 3(a2 + b2 ) and V y = -6a.

Lumps constitute a new phenomenon associated with the (2+ l j-dimensional easel.

They share an important property with one-dimensional solitons. Thus, the lump solu­

tions correspond to reflectionless potentials. However, they are potentials transparent

at one fixed energy while the one-dimensional solitons are potentials transparent at all

values of the energy.

Solutions of (2 + I)-dimensional integrable equations which depend on arbitrary

functional parameters constitute another new and interesting class. Such solutions

of integrable equations representable in the form (1.3.1) were first found within the

framework of the dressing method [24] . In order to construct such solutions it is sufficient

to consider the factorizable kernel of the operator P: F(x, z,y, t) = <p(x, y, t)rp(z,y, t).

In this case, the initial value <p(x ,0, 0) of the function sp is arbitrary. Accordingly, the

dressing method proposed in [24] is important for construction of classes of solutions for

(2 + I)-dimensional integrable equations. Moreover, the dressing method also admits

generalization to integrable equations connected with pairs of operators £1, L2 of the

form L, = Ai (8:r2)8:r; + Bi (8:r3) (i = 1,2) [260].

t It is noted that that notion of lumps was first introduced in a more general

framework by Coleman [281].

36 Chapter 1

The Hirota direct method (see, for example, [48,49]) is also a powerful method for

the construction of exact solutions to (2+ I)-dimensional integrable equations. Indeed,

the multisoliton solutions of the KP equation were first constructed by just this method

[282].

The construction of the exact solutions is one of the most important aspects of

the theory of the integrable equations. Solutions which are free from singularities are

naturally of special interest in connection with physical applications.

In general, the dressing method is not best suited to the solution of Cauchy

initial value problems for nonlinear evolution equations. However, in one particular

case, namely, that of the three-wave system (1.3.9), the introduction of characteristic

coordinates allows one to solve completely the inverse problem for the auxiliary system

(1.3.11) in a standard manner . The Cauchy initial value problem for the system (1.3.10)

may then be solved [256].

In general, higher-dimensional scattering problems are essentially different from

one-dimensional problems within the framework of the 1ST method. Thus, the eigen­

values of the partial differential operators L 1 playa different role to that of the spec­

tral parameter in the one-dimensional case. Indeed, for operators L 1 of the form

L1 = a8y+u(8x), the problem L11/Je = e1/Je may be converted into the problem L 11/J = 0

by the change 1/Je = eey/u1/J so that the parameter e is eliminated [90]. On the other

hand, for integrable equations which possess the triad representation (1.3.21), such as

the NVN equation (1.3.22), it is necessary to consider the problem L11/Je = e1/Je at one

fixed value of e. Further, in the multidimensional case, the inverse problem data arehighly overdetermined so that the problem of their characterization arises [262, 283J. Allthe features alluded to above complicate appreciably the analysis of multidimensionalinverse problems.

In order to formulate and solve the inverse problem for two-dimensional scattering

problems, it is necessary to introduce an auxiliary spectral variable in some manner

appropriate to the problem under consideration. This variable should playa role similar

to that of the spectral parameter in the one-dimensional case.

Let us consider, by way of illustration, the KP-I equation with potential u(x, y, t)

decreasing at infinity [284] . In this case, the spectral problem Ll1/J = (i8y + 8; +u(x, y, t))1/J = 0 is the one-dimensional nonstationary Schrodinger equation. The scatter­

ing problem is formulated in a standard manner. Thus , one introduces the two solutions

F+(x,y,tj A) and F-(X,y,t jA) which have a simple behavior as y -+ ±oo, F±(X,y,tjA)

--+ exp(iAx - iA2 y) where A is a real parameter (the momentum of the plane wave).y-+±oo

The decomposition of one complete basis {F+(X,y ,t jA), -00 < A < oo} with respect

to the complete basis {F-(x, y, t j A), -00 < A< oo} gives a scattering matrix S(N, A) :

/

+00F+(x, y, t j A) = F- (x , y, tj A) + -00 dA'F- (x, y, t j N)S(A', A, t). (1.4.2)

Introduction 37

In contrast to the one-dimensional case, the scattering matrix 8 depends now on the

two variables >'" and >... For real potential u, the matrix 1 + 8 is unitary. Then the

equation L2tP = 0 where the operator L2 is given by (1.3.3), shows that

d8(~~>.., t) = 4i(>..3 _ >..13)8(>"', x, t).

The latter equation is readily integrated to give [284]

8(>..',>..,t) = 8(>..',>..,O)exp4i(>..3 - >..,3)t.

(1.4.3)

It is seen that the mapping u(x, y, t) -+ 8(>"',>.., t) produces a linearization pro­

cedure in complete analogy with the one-dimensional case. Specifically, if u(x, y, t)

evolves according to the KP-I equation then the scattering matrix 8(>"',>.., t) evolves

according to the linear law (1.4.3). There is, however, the essential difference that in

the two-dimensional case the elements of the scattering matrix are not the inverse data

through which the potential u is reconstructed [284]. Accordingly, the introduction of

the spectral variable via the solutions F±(x,y, tj >") must be discarded.

There exists, however, another possibility. This corresponds to reformulation of the

auxiliary linear problems for the KP equation, namely, the equations LitP = 0 where the

operators Ll and L2 are given by (1.3.3), in such a way that they contain the required

spectral variable explicitly. This can be achieved, for instance, by consideration of

solutions of the linear problems of the form tP = X(x, y, >") x ei AX+ ~: Y, where>.. is an

arbitrary complex variable [284] . The auxiliary linear system for the KP equation is

thereby converted into the following system:

- 2 ·LIX = (a8y + 8x + 2z>"8x + u(x, y,t))X = 0,

£2X = (8t + 48; + 12i>"8; - 12>..28x + 6u8x

+ 3i>"u + 3iu8x + 3ux - 3a(8;luy))X = O.

(1.4.4a)

(1.4.4b)

Since_ . \ ,2

L -'''"'X-......"Li = e fT i. \ ~e'''"'x+ tr y (i = 1,2),

it follows that the condition [£1, £2] = 0 is evidently equivalent to the KP equation.

This turns out to be the appropriate way to introduce the spectral parameter in the

present context. In particular, for the KP-I equation it allows the construction of the

solution of the linear problem (1.4.4a) with the desired analytic properties in >.. [184] .

Thus, let us consider solutions X+ and X- of equation (1.4.4a) which are bounded

for all complex >.., tend to unity as >.. -+ 00, and obey the integral equations

X±(x, y, >") = 1 - f f dx' dy' G±(x - x' ,u- y', >..)u(x' , y')X±(x', y', >..), (1.4.5)

38 Chapter 1

where G± are Green functions for the operator Lo = i8y+8; +2i)"8x • Green functions

G+ and G- can be constructed which are analytic for 1m).. > 0 and 1m).. < 0, respectively

[284J. Consequently, the corresponding solutions X+ and X- of equations (1.4.5) are

meromorphic in the upper and lower half planes 1m).. > 0 and 1m).. < 0, respectively. The

existence of poles for the functions X+,X- is connected with the existence of nontrivial

solutions Xf for the homogeneous integral equations (1.4.5). Multidimensional Green

functions, similar to G±, and, indeed , problems of the type (1.4.4a) first appeared in

work of Faddeev [285] (see also [283]) devoted to the construction of transformation

operators for the multidimensional Schrodinger equation.

It follows from equations (1.4.5) that the boundary values of the functions X+ and

X- on the real axis are connected by the relation [284]

(1m).. = 0) (1.4.6)

for appropriate :F{)..', )..) . Similar formulae hold for the functions 1/J± = x±ei >,x - i >.2y •

As a consequence, we arrive at a conjugation problem, namely, a Riemann-Hilbert

problem for the construction of the function X{)..) which is analytic for 1m).. ::f 0 and the

boundary values of which on the real axis 1m).. = 0 are connected by the relation (1.4.6).

In contrast to the local problem (1.2.12), the relation (1.4.6) is nonlocal. Accordingly,

it is quite natural to refer to the corresponding conjugation problem as the nonlocal

Riemann-Hilbert problem [284].

The solution of the nonlocal Riemann-Hilbert problem is given by singular linear

integral equations. The function :F{)..',)..) corresponds to the inverse problem data while

u{x, y, t) is given by

u{x, u, t) = .!. 88 F" r+ood)'" d)" :F{>.', )..)x- (x, u, t, )..)ei (>" _>.)X_i(>.'2_>.2)y

1r x L; L;

The poles of the functions X±(1/J±) correspond to the lumps of the KP-I equation. Equa­

tion (1.4.4b) determines the evolution of the inverse problem data F{)..' ,).., t) in time:

:F{)..' ,).., t) = l{)..',).., 0) exp 4i{)..13 - )..3)t. As a result, the initial value problem for the

KP-I equation can be solved, in principle, by the standard form of the 1ST method

u(x, y, 0) .-!..{F()..',)..,0), . . .} ..!l{F()..',).., t) , . . .} !!! u{x, y, t) .

This observation of Manakov, that in the two-dimensional case a nonlocal Riemann­

Hilbert problem arises rather than a local one, has played a significant role in the

development of the two-dimensional 1ST method.

Introduction 39

The general nonlocal Riemann-Hilbert problem may be formulated as follows [270,

2711 . Let I' be a contour on the >.-complex plane and let the matrix-valued function

T(>.', >') of two variables be given on the tensor product r ~ r . The nonlocal Riemann­

Hilbert problem involves the construction outside r of an analytic function X(>') whose

boundary values on I' are connected by the relation

(>' E r). (1.4.7)

The uniqueness of the solution to the nonlocal Riemann-Hilbert problem is achieved by

an appropriate normalization. Similar to the local case, the normalization X(00) = 1 is

regarded as the canonical one. As in the local case the function X(>') outside I' is given

by the formula (1.2.13) while the jump k(>') is determined from the equation

k(>') = rd>.'T(>.', >') +~ rrd>.' d>'" k~>'")~~>'" ~)l- 2m Jr l- >. - >. + zO(1.4.8)

The latter follows from (1.4.7) after the substitution of the relation (1.2.14).

The formulae (1.4.8) and (1.2.13) give the solution ofthe regular nonlocal Riemann­

Hilbert problem. Equation (1.2.14) with k(>') = frd>.' X- (>.')T (>.' ,>') and formula

(1.2.13) also give the solution of the nonlocal Riemann-Hilbert problem. In order

to accommodate the lump solutions it is necessary to consider the singular nonlocal

Riemann-Hilbert problem, namely the problem of construction of the meromorphic

function which obeys the relation (1.4.7) .

The nonlocal Riemann-Hilbert problem possesses explicit solutions which contain

functional parameters [2711. Such solutions arise in the case of a degenerate kernel T of

the form T(>.' ,>') = E~=l fn().')gn().) . In this case, equation (1.4.8) is converted intoa system of N linear algebraic equations

N

~n + L ~mAmn = hn ,m=1

where

N

and k(>') = L ~ngn(>') . The formula (1.2.13) gives solutions of the nonlocal Riemann-n =1

Hilbert problem dependent on arbitrary functions. It is noted that such solutions are

absent for the local Riemann-Hilbert problem which corresponds to the case T().' ,).) =

t5(>.' - >')T(>').

40 Chapter 1

As in the local case, the nonlocal Riemann-Hilbert problem is formulated in the

space of the auxiliary spectral variable >.. It is thus well-adapted to the solutions of

integrable equations which contain all independent variables on an equal footing. A

broad class of solutions can be constructed on choice of an appropriate contour r.

If one of the variables, x, say, is distinguished and contour r is the real axis,

the nonlocal Riemann-Hilbert problem is equivalent to the factorization problem for

the integral operator on the x-axis. Indeed, on multiplication of equation (1.4.8) by

exp i>.(z - z), integration over >. along the real axis, and introduction of the functions

1 /+00/+00F(x, z) = - d>.d>.'T(>.', >.)ei)'(x-z),211" -00 -00

K(x, z) = --21 /+00 ou»; x)ei)'(x-z),11" -00

one obtains the celebrated Gelfand-Levitan-Marchenko equation (1.2.6a).

Thus, the method of solution of the nonlinear integrable equations which employs

the nonlocal Riemann-Hilbert problem formalism is more general than the original

dressing method based on integral operator factorization. The nonlocal Riemann­

Hilbert problem approach works for a wide class of the (2 + I)-dimensional integrable

equations. However, this procedure is not general enough to cover all known integrable

equations in 2 + 1 dimensions. Thus , it is recalled that, in general, the formulation

of the inverse problem as a Riemann-Hilbert problem is predicated on the existence of

sectionally-meromorphic solutions of the given spectral problem with jumps on some

contour r. For one-dimensional spectral problems, it seems that such solutions always

exist.

However, a quite different situation arises in the two-dimensional case. The DP-II

equations (1.3.2) (a = 1) provide the simplest example. In this case, we again have the

equations (1.4.4) and (1.4.5) but now with the change i8y -+ ay • This crucially changes

the properties of the solutions of equations (1.4.4). The Green function of the operator

Lo = ay + a;+ 2i>.ax and, consequently, the solutions Xof the corresponding integral

equation (1.4.5) are nowhere analytic in >. and have no jump across the real axis. Hence

the conjugation problem cannot be formulated.

The way around this problem for the KP-II equation was given by Ablowitz, Fokas,

and Bar Yaacov [286] . It turns out that there is no need to consider the solutions with

restrictive analytic properties. It is sufficient to consider the solution X(x, y, t, >') of

equation (1.4.4a) (a = 1) which is bounded for all >. and then calculate the derivative

aX/a>.. The use of an integral equations of the type (1.4.5) together with properties of

Introduction 41

the corresponding bounded Green function and the symmetry property of the solution

X allows one to perform such calculation. For the KP-II equation one has [286]

(1.4.9)

where :F(>., X) = F(>'R, >'1) is an appropriate integral expression over u(x, y, t) and

X(x,y, t) and >'R = Re>., >'1 = Imx. It is noted that the function 1/J = Xexp(i>'x + >.2 y)

obeys the equations 81/J(>')18X = :F(>'))1/J(-X).

One now requires the following generalization of the integral Cauchy formula (see,

for example, [287])

1 II -8xI8ji 1 1 X(JL)x(>') = -2. dJL /\ dJL--, + -2' dJL--, ,1l"t JL - /I 1l"t &r! JL - /I

n

(1.4.10)

where n is a region on the complex plane, 8n is its boundary, and dJL/\dji = -2idJLRdJLI.

On use of formula (1.4.10) in the concrete case (1.4.9), with X -+ 1 at >. -+ 00 and with

the full complex plane C and n , one obtains

1 II -:F(JL, Jl) X(x, y, _ji)e-i (2/-tRX+ 4/-t R/-tIY)X(x, y, >') = 1+ -2. dJL /\ dJL >. .

m JL-e

In addition one has [286]

u(x, y, t) = ;: :x II -d>'/\ dA:F(>., A)X(x, y, _A)e-i (2ARx +4ARA, y ) .

e

(1.4.11)

(1.4.12)

Equation (1.4.11) together with formula (1.4.12) provide the solution of the inverse

problem for the KP-II equation. The function :F(>.,X) corresponds to the inverse

problem data. The evolution of t he inverse problem data :F(>., A, t) in time is determined

by equation (1.4.4b) with a = 1. It is of the form F(>. ,X, t) = :F(>., A, 0) exp( -4i(>.3 +A3)t). This allows us to solve the initial value problem for the KP-II equation in a

standard manner.

The so-called 8-equation (1.4.9) plays a pivotal role in the approach described above

and indeed, has given the name 8-method to the procedure proposed in [286, 288]. The

8-equation provides the equations of the inverse problem for the KP-II equation. It is

crucial that the right hand side of the 8-equation (1.4.9) is linear with respect to X.

Only in that case, are the equations of the inverse problem which follow from (1.4.10)

linear singular integral equations.

The connection between the 8-equation and the 1ST method was first discovered

by Beals and Coifman [289] . Thus, in a study of the one-dimensional matrix spectral

problem, they noted that one can treat the corresponding Riemann-Hilbert problem as

42 Chapter 1

a particular case of the a-problem. However, in the one-dimensional case, introduction

of the a-problem is not a necessity.

The discovery by Ablowitz, Fokas, and Bar Yaacov that the a-method [286] was

essential to the treatment of two-dimensional problems has proved of fundamental

importance in the subsequent development of the higher-dimensional version of the

1ST method.

A number of the (2+ I)-dimensional nonlinear integrable equations such as KP-II,

DS-I, NVN-I equations have been treated by the a-method. The main stages in the

procedure are the same as that for the KP-II equations [288] . Thus, it is first necessary

to define the eigenfunctions X(x,y, >') of the spectral problem L1x = 0 which are

bounded for all >. and appropriately normalized. Usually such functions are determined

as solutions of Fredholm integral equations of the second type. Nontrivial solutions of

the corresponding homogeneous Fredholm equation lead to lump solutions. Next, one

must calculate 8X(>.)/8Xj this derivative is usually expressed via the inverse problem

data F(>.,X, t) and another solution N(x, y , >') of the initial spectral problem. Then the

interrelation between the functions N(x, y , >') and X(x, y, >') must be determined. This

relation is a linear one in all known cases and as a consequence one obtains the linear a­equation. With the aid of formula (1.4.10) this linear a--equation generates the inverse

problem equations. It is further required that one should express the potential u(x, y, t)

via the inverse problem data and the function X(x,y, >.) . Finally, the second auxiliary

linear problem L2X = 0 determines the time-dependence of the inverse problem data

{F(>', >.', t), .. .}. As a result, one is able to solve the initial value problem for the given

nonlinear integrable equation via the standard scheme

u(x, u, 0) .!.{F(>., >.',0) , . . .}E.{F(>., >.' , t), . . .}!!! u(x ,y, t).

It is noted that the overall procedure is the same as the corresponding procedure (1.1.4)

in 1 + 1 dimensions. The main technical difference is that now the stages I and III are

connected with the a-problem.

At the present time, the a-method represents the most general version of the 1ST

method and the a-equation is the most general generator of inverse problem equations

[271, 288, 290-294].

The general linear matrix a-problem is formulated as the following nonlocal 8­problem [271]

8X~i X) = (X*R)(>., X) = JJd>.' 1\ dX'X(>", X')R(>.', X'; >., X),c

(1.4.13)

Introduction 43

where R(>.', ~'j >.,~) is a matrix-valued function. For the singularity free function Xwith

the canonical normalization X(oo) = 1, the a-equation (1.4.13) is equivalent, by virtue

of (1.4.10), to the integral equation

x( >. ~) = 1+ _1 lid>" A d~'(x *R)(>.', ~')'21l"i >.' - >. .

c

(1.4.14)

On substitution of (1.4.14) into (1.4.13), one obtains the following equation for k(>') :=

a~ [271]f)>'

k(>',~) = II d>.' Ad~'R(>",~'j>'))+c

_1 II lid Ad- d>.' Ad~,k(J.L ,p)R(>.',~';J.L,p)21l"i J.L J.L. J.L _ >.' .

c c(1.4.15)

Equation (1.4.14) or equation (1.4.15) gives the solution of the general nonlocal a­problem (1.4.13) with canonical normalization.

Equations (1.4.14) and (1.4.15) are two-dimensional singular integral equations with

independent variables >'R and >'1 . Let us compare these equations with the equations

(1.2.13), (1.2.14), and (1.2.15) which solve the local Riemann-Hilbert problem. Their

main common feature is that they both are singular linear integral equations. The

formulation ofthe inverse problem equations in the form (1.4.14) is a logical development

of the treatment of inverse problem equations via singular integral equations. This

approach was first proposed in connection with the one-dimensional NLS equation [6) .

The basic difference between equations (1.4.14) and equations (1.2.14) is that equation(1.2.14) which solves the local Riemann-Hilbert problem is a one-dimensional equation

(>. E I') while equation (1.4.14) which solves the nonlocal a-problem (1.4.13) is a two­

dimensional integral equation. Equations of the type (1.2.14) which solve the nonlocal

Riemann-Hilbert problem occupy an intermediate position between them.

The nonlocal a-problem (1.4.13) contains the problems discussed above as par­

ticular cases. The simplest example corresponds to the local function R = 15(>" - >.)15(~' - ~)R(>', ~) . In this case, the problem (1.4.13) is reduced to the local a-problem

ax(~, ~) = (>. ~)R(>' ~)a>. x, , . (1.4.16)

In the case R = 15(>.' - ~(>., ~))15(~' - ~(>., ~))R(>', ~), where ~(>.,~) is an appropriate

function, one obtains the quasi-local a-problem or the a-problem with shift

(1.4.17)

44 Chapter 1

For instance, the a-equation (1.4.9) corresponds to the case ~ = -X. Finally, if

R(A', X'; A, X) = 6r(A').R(A',A)6r(A), where 6r(A) is the Dirac delta-function concen­

trated on r, then a~ = X+-X- on the contour r and the nonlocal a-problem (1.4.13) is

reduced to the noJt~calRiemann-Hilbert problem (1.4.7). The general integral equation

(1.4.14) is reduced under these restrictions in a corresponding manner. For concrete

two-dimensional problems, the function R(A',X'jA,X) usually plays the role of inverse

problem data for a class of decreasing potentials. In these cases R is, in effect, a function

of two real variables.

The various special cases of the general a-problem mentioned above correspond to

different (2+ I)-dimensional integrable equations. The inverse problem is formulated as

a pure nonlocal Riemann-Hilbert problem in the case of the KP-I [284, 295] and OS-I

[296] equations or as the pure quasi-local a-problem (1.4.17) for the KP-II [286] and

OS-I [296] equations. In other cases (such as the NVN-I equation [297]) the inverse

problem is a composition of the nonlocal conjugation problem and the quasi-local a­problem. The lump solutions correspond to solutions X(A) with poles and are calculated

via purely algebraic equations. They have been constructed for the KP-I, OS-I, NVN-I,

and other (2 + I)-dimensional integrable equations.

The general nonlocal a-problem (1.4.13) can also be set in the context of a general

version of the dressing method [271, 298, 299] . This a-dressing procedure allows us

to construct a wide class of compatible multidimensional integrable systems which

represent broad generalizations of known integrable equations [271, 298, 299].

The a -approach changes in an essential way the analytic view of the 1ST method

even in (1 + I)-dimensional problems. Thus, reformulation of one-dimensional inverseproblems in the shape of a a-problem allows both the construction of more generalsolutions and a deeper understanding of the structure of integrable equations [271, 300­307].

The a-approach also sheds light on the deep interrelation between the 1ST method

and the theory of complex variables. The 1ST method in the a-formulation turns

out [308, 309] to be deeply connected with the theory of generalized (pseudo) analytic

functions introduced by Bers [310] and Vekua [311] . Such generalized analytic functions

obey generalized Cauchy-Riemann equations

a~\A) = A(A)x(A)+ B(A)X(A), (1.4.18)

where A(A) and B(A) are scalar functions. There exists an established theory of

generalized analytic functions which represents an extension of the classical theory of

analytic functions and preserves many of its important features [310-313].

Comparison of (1.4.18) and (1.4.16) shows that the solution of the local a -problem

and also of the a-problem which is reducible to the form (1.4.18) (for example, the

Introduction 45

KP-II and NVN-I equations) are nothing but generalized analytic functions. This fact

together with properties of the generalized analytic functions allows us to considerably

strengthen certain results in inverse problem theory [308, 309).

Finally, within the a-formulation of the 1ST method, much symmetry is revealed

between the initial spectral problem with potential u(x, y) (for example, (8y +8; +2iA8x+u(x, y))X =0) and the equation which contains the inverseproblem data F(A,A),

namely the a-equation. Indeed, the a-equation :~ = RX is merely the two-dimensional

linear problem with Green's function G = ~ (AI ~ A)' the use of which allows us to

rewrite the 8-equation in the integral form (1.4.14). This symmetry becomes more

evident if one rewrites the local a-equation :~ = R(A)X(A) + Q(A)X(A) in the 2 x 2matrix form

(8~n + A 8~I + W(An , AI)) <P = 0, (1.4.19)

where A = (~ ~i)' Q = (X, X)T, and W = -2 (Q ~) . The form (1.4.19) of the

a-equation demonstrates the duality between the independent variables x , y and the

spectral variables An, A}, the potential u(x, y) and the inverse problem data R(An, AI)

and, finally, between the spectral problems in the initial variables x, y and in the spectral

variables An,A} . This manifest duality is one of the most interesting features of the a­approach. Similar types of duality may arise for other classes of the spectral problems.

Thus, dual spectral problems different from the a-equation (1.4.19) may exist. A study

of such cases would be of interest in the theory both of inverse spectral transforms and

of associated nonlinear integrable equations.

This completes our introductory survey. In the sequel, we proceed to the main con­tent. In the next chapter, the a-method and the nonlocal Riemann-Hilbert procedure

is applied to (2+ I)-dimensional integrable equations. In Chapter 3 other methods and

integrable equations are described. General methods for the construction of the (2+ 1)­

dimensional equations (the T-function method, and a-dressing method) are discussedin Chapter 4. Chapter 5 is devoted to multidimensional integrable systems.

Chapter 2The Inverse Spectral Transform Method

in 2+1 Dimensions

In this chapter, we will consider three basic examples of integrable equations in 2+ 1

dimensions. These are the Kadomtsev-Petviashvili (KP) equation, Davey-Stewartson

(DS) equation, and Nizhnik-Veselov-Novikov (NVN) equation. Most attention will

be devoted to methods of solution of these equations, namely, to associated nonlocal

Riemann-Hilbert problems and the a-method. We also discuss special classes of solution

to these equations such as lumps, soliton-type solutions, as well as solutions dependent

on functional parameters.

2.1. The Kadomtsev-Petviashvili-I equation

We start with the KP equation

(2.1.1)

This is probably the most studied of nonlinear integrable equations in three independentvariables x, y, t. It describes a slow variation in the y-direction of wave propagation in the

x-direction [248]. The KP equation (2.1.1) is equivalent to the compatibilit.y condition

for the system [24, 249]

L 1'l/J = (O-oy + a; + u(x,y, t))'l/J = 0,

L2'l/J = (at + 4a~ + 6uax + 3ux - 3a(a;I U y ) + a)'l/J = 0,

(2.1.2a)

(2.1.2b)

where (a;IJ)(x,y):= [Xoo

dx'f(x',y) and a is an arbitrary constant.

The KP-I equations (a = i) and the KP-II equation (a = 1) are transformed into

each other by the simple change y -+ iy. However, the methods of solution of these

equations are essentially different. This is connected with the distinction between the

properties of the linear equation

(aay + a; + u(x, y))'l/J = °47

(2.1.3)

48 Chapter 2

in the case a = i (the one-dimensional nonstationary Schrodinger equation) and in the

case a = 1 (the one-dimensional heat equation).

First, let us consider the KP-I equation. We will assume that the scalar function

u(x ,y, t) decreases sufficiently rapidly as x2+ y2 -+ 00 and that 1:00dxu(x,y, t) = O.

We shall follow, in the main, the papers [284, 295J .

We first seek to solve the inverse problem for equation (2.1.3). In this connection,

we introduce the spectral variable>. by consideration of solutions 'l/J of equations (2.1.2)

of the form 'l/J(x, y, >') := X(x,y, >') exp(+i>'x - i>.2y), that is, via the transition from

equation (2.1.3) to equation (1.4.4a). For the KP-I equation (a = i) it adopts the form

(i8y+ 8; + 2i>'8x + u(x ,y,t))X = O. (2.1.4)

The major part of this section will be devoted to the investigation of spectral problems

associated with (2.1.4).

First, we note that equation (2.1.4) with u == 0 admits solutions XO that can be

represented in the form

xo(x, y, >') =1:00dkA(>', k)eikx- i(k2+2k>')y = [:00 d>.'B(>. ,>.')ei(>" _>')X_i(>.'2_>.2)y,

(2.1.5)

where A(>., k) and B(>., >.') are arbitrary functions. Particular solutions of the type

(2.1.5), namely, xo = 1 and xo(x, y, >., JL) = ei(p.->,)x-i(p.2_>.2)y, will arise many times in

the subsequent discussion.

It proves convenient at the outset to convert the partial differential equation (2.1.4)

to the equivalent integral equation

x(x,y,>.) = xo(x,y,>.) - (Gux)(x,y) ,

where, in the general case, the free term XO is of the form (2.1.5)

1+00 1+00

(GJ)(x,y) := -00 -00 dx'dy'G(x-x',y-y',>')!(x',y')

(2.1.6)

(2.1.7)

and the Green function G(x - x' ,y - y',>.) is defined in the standard manner, namely,

via

LoG:= (i8y +8; +2i>'8x)G(x - x', y - y', >') = o(x - x')o(y - y') , (2.1.8)

where o(x) is the usual Dirac delta-function. The solutions of the integral equation

(2.1.6) which correspond to different choices of the free term XO and the Green function

G determine the different classes of solutions of the partial differential equation (2.1.4).

Inverse Spectral Transform Method in 2 + 1 Dimensions

The formal expression for the Green function G(x, y, >.) is given by

1 /+00 /+00 eikx+ik'yG(x, y, >.) = - (27r)2 -00 -00 dkdk' k2+ 2k>. + k"

49

(2.1.9)

It is readily seen that the function G(x, y, t) is not defined for all >. = >'R + i>'l. Indeed,

for real >., the integrand in (2.1.9) has poles on the real axis and , hence, the function

G is ambiguous for >'1 = O. As a result, the function G has a jump across the real axis

>'1 = O.

For complex>. (>'1 #- 0) , on integration in (2.1.9) over k', we obtain

(2.1.10)

where O(~) is the Heaviside funct ion:

O(~) = {I, ~ > 00, ~ < 0

Hence, the Green function G is well-defined for complex>. at >'1 #- O. The boundary

values G+, G- of this function at >'1 -+ ±O follow from (2.1.10) and are given by

G+(x,y,>.) = ~ /+00 dk(O(y)O(-k) _O(_y)O(k))eikx-i(k2+2>'k)y27rZ -00

together with

tr t«,u, >.) = -21.1+00

dk(O(y)lJ(k) -O(-y)O(_k))eikx-i(k2+2>'k)Y.

7rZ -00

(2.1.11)

(2.1.12)

The functions G+ and G- admit analytic continuation to the upper (>'1 > 0) and lower

(>'1 < 0) half-planes, respectively. On the real axis >'1 = 0, one has

(2.1.13)

where sgn ({) := O({) -O(-~) Note that the right hand side of (2.1.13) is a solution of

the type (2.1.5) of equation (2.1.4) with u == 0 since Lo(G+ - G-) = O.

The Green functions G+ and G- are just those which allow the construction of

solutions of equation (2.1.4) with good analytic properties.

Let us now consider functions x+(x, y, >.) and x- (x, y,>.) which are bounded for all

complex >. = >'R + i>'I, tend to 1 at >. -+ 00, and which are, in turn, solutions of the

integral equations

(2.1.14)

50

and

Chapter 2

(2.1.15)

where the Green functions G+ and G- are given by the (2.1.11) and (2.1.12) , respec­

tively. Let us introduce the function

( \) ._ {x+(X, y, A), AI > 0,Xx,y,'" .-

X-(X,y,A), AI < 0.

This is a solution of equation (2.1.14) bounded for all A and X ~ 1 at A~ 00.

To ascertain the propert ies of the function X we must investigate the integral equa­

tions (2.1.14) and (2.1.15) . These are Fredholm integral equations of the second type

with a parametric dependence on A. We will assume that the kernels G±(x,y,A)U(X, y)

in (2.1.14) and (2.1.15) are such that standard Fredholm theory can be applied.

For convenience, we recall some elements of the theory of the Fredholm integral

equations

x(~) = XO(~) - (FX)(~) (2.1.16)

of the second type (see, for example, [314, 315]). The Fredholm determinant ~ is

one of the most important concepts underlying this theory. Formally, it is defined by

ln~ = tr(ln(1 + F) - F) where tr denotes the operator trace. If ~ '" 0, then equation

(2.1.16) is solvable for all xo(~). In the case ~ = 0, the homogeneous equation (2.1.16)

may have nontrivial solutions and the properties of the inhomogeneousequation (2.1.16)

are the subject of the well-established Fredholm theorems [314, 315). In particular, for

the solvability of equation (2.1.16) in the case ~ = 0, the free term XO in (2.1.16) must

be orthogonal to all solutions Xi of the equation which is adjoint to the homogeneous

equation (2.1.16).

In our case, formally X±(A) = (1 + a±u)- . 1. The functions G± are analytic at

AI > 0 and AI< O. Hence, the functions X+(A) and X-(A) are also analytic in the upper

and lower half-planes respectively with the exception of the points where the Fredholm

determinant of the operator 1 + c-« vanish. At these points Ai (with ~±(Ai) = 0),

the functions X+ and X- have poles. Thus, the existence of nontrivial solutions of the

homogeneousequations (2.1.14) and (2.1.15) is connected with the presence of poles for

the functions X+ and X- . The investigations of properties of the Fredholm determinant

is rather complicated. Here, we assume that the Fredholm determinants ~±(A) for

equations (2.1.14) and (2.1.15) have a finite number of simple zeros at the points Atand none of them are situated on the real axis AI = 0. The corresponding solutions

of the homogeneous equations (2.1.14) and (2.1.15) will be denoted by xi(x,y) and

xi(x,y) .

Inverse Spectral Transform Method in 2 + 1 Dimensions 51

Accordingly, Fredholm theory implies that the solutions of equations (2.1.14) and

(2.1.15) are of the form

±( ') 1 .~ xt(x, y) A±( ')X x,y,/\ = +~L.." A-A* +X x,y,/\,i=1 '

(2.1.17)

(2.1.18)

(2.1.19)

where X+(X,y,A) and X-(X,y,A) are analytic functions in IffiA > 0 and ImA < 0

respectively. The functions xt(x, y) are normalized such that xt(x - 2Aty) --+ 1 as

x2+ y2 --+ 00. For real potentials u(x ,y), one has Ai = At, where the bar indicates

complex conjugation.

The functions xt satisfy the following important relation [295]

lim (x± - i, xt ±) = (x - 2A;y + "It(t))xtA-At /\ - Ai

and

1 =f 2~1:001:00

dxdyu(x,y)xt(x,y) = 0,

where "It are time-dependent constants.

To establish these relations, for instance, for the function X- we introduce the

auxiliary functions

(2.1.20)

(2.1.21)

and

Xi(X, y, A) := xi(x, y)ei>,x - iA2

y .

It follows from equation (2.1.15) that

«1 + (;-u)J.ti{-, A))(x, y) = ei>.x - i>.2y - A_i A:- «1 + 6-U)Xi(" A))(x, y), (2.1.22)

•

where the function 6- is of the form

Passing to the limit A --+ Ai in (2.1.22) and taking into account that

(2.1.24)

one obtains

(2.1.25)

52

Hence,

and

1 /+00/+001+ 211" -00 -00 de dT/u(~, T/)xi(~, T/) = 0

_( \_) .aXi(x,y,~n A ( \_)

Iti X,y'''i +~ a~ =,iXi X,y'''i ,

Chapter 2

(2.1.26)

where ,i are constants. Finally, it is not difficult to show that equation (2.1.26) is

equivalent to equation (2.1.18) for X-. The relations (2.1.18) and (2.1.19) for the

function X+ are proved analogously.

Thus, in view of (2.1.17), the problem of construction of the solution of equation

(2.1.4) with good analytic properties is solved: the function X(A) = {X~' ~I > 00 isX , ",<

analytic on the entire complex plane A except at a finite number of simple poles and

has a jump across the real axis AI = O. It is the existence of this jump that allows us to

formulate the corresponding conjugation problem and introduce inverse problem data.

With this in mind let us calculate the quantity K(X,yjA) := X+(X,y,A) - X-(X,y,A)

for real A. On subtraction of equations (2.1.14) and (2.1.15), we obtain

K(x, Yj A) = ((G- - G+)UX+)(x, y, A) - (G-(-, A)u(·)K(·;A)) (x, y).

Use of (2.1.13) then gives

r+OO

K(x, Yi >') = J-oo d>.'T(>., >.')eW(A' ,A,X,y) - (C-(', >.)u(·)K(· ;>.»(x,Y)

where

W(A',A,X,y) := i(A' - A)X - i(A,2 - A2)y

and

i /+ooJ+ooT(~, A') = 211" sgn (A - A') -00 -00 de dT/ u(~, T/)x+ (C T/, A)eW(A,A' ,e,'I/).

(2.1.27)

(2.1.28)

(2.1.29)

(2.1.30)

Accordingly, K(x, y, A) satisfies an integral equation of the type (2.1.15) with free

J+ OO

term -00 dA'T(A, N)eW(A' ,A,X,y). In view of this, it is natural to consider solutions

N(x, y, AI,~) of equation (2.1.4) which are simultaneously solutions of the integral

equation

(2.1.31)

where ~I = A~ = O. The solutions N(x, y, N, A) with different AI form a one-parameter

family of solutions of equation (2.1.4).

Inverse Spectral Transform Method in 2 + 1 Dimensions 53

If we now multiply equation (2.1.31) by T(>. ,>.') , integrate over >.', and then

compare the resulting equation with equation (2.1.28) , then we obtain

1+ 00

K(x,yj>') = x+(x,y,>.) - x-(x,y,>.) = -00 d>.'T(>.,>.')N(x,y,>.' ,>.). (2.1.32)

It is assumed that the homogeneous equation (2.1.15) has no nontrivial solut ions for

real x.Further, in order to formulate (2.1.32) as a relation between boundary values of

a meromorphic function, one must find the interrelation between N(x, y, >.', >') and

X-(x,y,>.). To do this, we introduce functions

N(x y x >') '= N(x y >" >.)eiAX-iA2y, , , . , , , , (2.1.33)

(2.1.34)

which, as follows from (2.1.31) and (2.1.15), are solutions of the integral equations

and

((1 + Gu)N(., >.', >.))(x,y) = eiA'x- iA,2y (2.1.35)

(2.1.36)

(2.1.37)

with the function G(x, y, >') being given by (2.1.33) . Note t hat G(x, y, >') is nothing

but the Green function for the operator i8y+8; while N(x,y , >., >') and X±(x, y, >') are

solutions of the original equation (2.1.3).

Operation on (2.1.35) with 8/8>. yields

((1 + eu) 8N(~;" >') ) (x, y) = F()..', )..)ei Ax - i>.2y

where

1 1+001+00

F(>",>.) = 21Ti -00 -00 ~d'flU(e,,,,)N(e,,,,,>.',>.)· (2.1.38)

On comparison of (2.1.37) and (2.1.36), if it is assumed that the equation (1+Gu)M = 0

has no nontrivial solutions for real >., we obtain

(2.1.39)

(2.1.40)

Finally, integration of the equality (2.1.39) over>. on use of (2.1.33) , (2.1.34) and the

relation N(x, y, >., >') = X- (x, y , >') gives

N(x, y, >.',>') = X- (x,y,>.')ei(A' -A)x-i(A'2_A2)y

+ r dJ1.F(>.', J1.)X-(x, v.J1. )ei(P.-A)X-i(p.2_A2)y.) A'

54 Chapter 2

The formula (2.1.40) is crucial for subsequent constructions. By virtue of (2.1.40)

the relation (2.1.32) can be rewritten as [284-295]

where

A' 00

f(>"', >") := T(>..', >..)+8(>..->..') 1-00 dJLT(>.., JL)F(JL, >..') -8(>'" ->..) 1, dJL,T(>.. , JL)F(JL, >..').

(2.1.42)

The latter expression for f(>"', >") can be simplified if one takes into account the formulae

(2.1.30) and (2.1.38) whence, following a straightforward computation, we obtain

, sgn (>" - >"') 1+00 1+00,

f(>.. , >") = 211'i -00 -00 dxdyu(x ,y)N(x,y,>.,>.), (2.1.43)

where 1m>.' = 1m>.. = O. The formula (2.1.43) manifestly demonstrates the significance

of the function N(x, y, >", >.) . Note that this function does not admit continuation from

real Xand >.'. Observe also that, for real potentials u(x ,y), one has J(>.',>') = f(>', >.').

Thus, our intermediate purpose has been achieved. We have constructed for equa­

tion (2.1.4) a solution X(x, y, >') which is a meromorphic function in >. in the upper

and lower half-planes and for which the boundary values X(x, y , >') are connected by

the relation (2.1.41). Hence, we have arrived at the nonlocal Riemann-Hilbert problem

described in Section 1.4. In our case, the contour r is the real axis, the function X

has the canonical normalization (X ----+ 1) and the function f(>.', >..) plays the role of theA-+OO

inverse problem data,

Note that for the functions 1/J±(>") which are defined by the formula (2.1.34) and

are solutions of equation (2.1.3), one has

(2.1.44)

instead of (2.1.41). For real >., the sets of solutions 1/J+(x,y,>..) and 1/J-(x,y,>..) form

independent complete bases in the space of solutions of equation (2.1.3) with a = i

[284].

The solution of the nonlocal Riemann-Hilbert problem is given by the formula

(1.2.13) on taking into account (2.1.17), that is, by the formula

n (+ _)x(x,y,>")=I+iL ~+~

. >.. - >.. . >.. - >..1=1 I I

1 1+00 1+00 ,x-(x, u,>.')f(>" ,JL)ew{A' ,IJ,X,y)+-2' .~ >. .

lI'l -00 -00 JL -(2.1.45)

Inverse Spectral Transform Method in 2 + 1 Dimensions 55

The equations associated with inverse problem now follow. Thus , passing in (2.1.45) to

the limit A- AR- iO, one obtains

(2.1.46)

where Im). = O. In the limits A- At, the formula (2.1.45) together with (2.1.18) yield

1 1+00 1+00 ,x-(x, y, N)f(N, IL, t)eW(>."J.l,x,y)+ -2. dud); ± (2.1.47)7l"~ -00 -00 IL - \

(i=I, .. . ,n) .

Equation (2.1.4) then gives directly

( t) - 2'1' ,8X(X,y,A)u x, y, - - ~ 1m 1\ 8 .>. .....00 x

(2.1.48)

On expansion of (2.1.46) into an asymptotic series over A-1, in view of (2.1.48), we

obtain

.(x,y,!) ~ ~ {2t<xi +X;)

+ ~ 1+00 1+00

dAdA'f(A', A, t)X- (x ,u, A')ei (>" _>')X_i(>.'2 _>.2)y} . (2.1.49)7l" -00 -00

Equations (2.1.46), (2.1.47), together with (2.1.49) constitute the system of equa­

tions which solve the inverse problem for equation (2.1.4) [295]. The set {f(N, A, t)

(ImA' = IffiA = 0), A;,X;, 'Yi(t) ,'Yi-(t), i = 1, .. . ,n } comprise the data of the inverse

problem. Given the latter, one can calculate the functions X- and xt (i = 1, .. . , n)

with the help of equations (2.1.46) and (2.1.47). Then one reconstructs the potential

u(x , y, t) via the formula (2.1.49) . Note that (2.1.46) - (2.1.49) are similar in form to

the corresponding equations (1.2.27) - (1.2.29) obtained for the KdV equation.

The nonlocal Riemann-Hilbert problem which generates the inverse problem equa­

tions is, as mentioned previously, a particular case of the a-problem. To provide a

concrete comparison between these two approaches we indicate here how equations

(2.1.46) - (2.1.47) can be derived within the framework of the a-method. For this

56 Chapter 2

purpose, let us consider solutions X of equation (2.1.4) bounded in Awhich also obey the

integral equation (2.1.6) with XO = 1. On use of the formula :x (A ~ AO) = 1l"l5(A-AO),

we obtain ~ = 0 for ImA '" 0 and ~ = G+ - G- for ImA = 0, where G+ - G- is

given by (2.1.13) . Then, on use of (2.1.17), one gets [288J

aX(X,y,A)=aX

n

i1r LXt6(A - An, ImA > 0i=1

ImA = 0,n

i1r L xi6(A - Ai), ImA < O.i = 1

(2.1.50)

Finally, taking into account (2.1.41) and (2.1.50) together with formula (1.4.10), we

retrieve precisely the formula (2.1.45) (dJ-t /\ dji, = -2idJ-tndJ-tI). Thus, we again arrive

at the inverse problem equations (2.1.46) and (2.1.47).

With the solution of the inverse problem for the spectral problem (2.1.4) com­

puted, we are now in position to address the integration of the KP-I equation. In

this connection, we next seek the temporal evolution of the inverse problem data

{f(A', A,t), At, Ai, 1't(t),1'i(t)}. The time dependence is determined by equation

(2.1.2b) or equation (1.4.4b) with (J = i. Indeed, applying the operator £2 with

a = -4iA3 to the relation (2.1.44) and taking into account that 1jJ± -. ei>,x- i>.2y as

x -. 00 and a;1 := J~oo' we obtain

(2.1.51)

To ascertain the time-dependence of At and 1';, it is sufficient to consider equation

(1.4.4b) at x, Y -. 00, that is, the equation

XL + 4Xxxx + 12iAXxx - 12A2Xx = O.

On substitution of the expression (2.1.17) into this equation, taking the limits A-. At,and using (2.1.18), we obtain

aAt _ 0 a1'f = 12(A:r)2at-'at " (2.1.52)

Hence, the temporal evolution of the inverse problem data is linear in accordance

to the basic feature of the 1ST method. As a result,

(2.1.53)

Inverse Spectral Transform Method in 2 + 1 Dimensions 57

The inverse problem equations (2.1.46) , (2.1.47), (2.1.49), and the formulae (2.1.53)

now allow us to solve the init ial value problem for the KP-I equat ion by t he standard

1ST method scheme [295]

U(X,y,D)-.!.{f(A',A,D), At(D), Ai(D) , 'Yt(D), 'Yi(Dn

E.{f(A' ,A, t), At(D), Ai (D) , 'Yt (t), 'Yi- (tn !.!! u(x, u,t).

Indeed, given u(x ,y ,D), solution of the forward problem for (2.1.4) provides the

data {f(A',A,D), A;(D) , 'Y;(Dn. For example,

i 1+001+00so: A, D) = 27l" sgn (A' - A) -00 -00 . ~ d77U(~ , 77 ,D)N(~, 77,D,A', A) .

Then, the formulae (2.1.53) give the inverse problem data for an arbitrary time t. On

use of these results and equations (2.1.46) and (2.1.47), we may, in principle , calculate

the functions x- (x,y, t , A) and xi (x,y,t), xi (x,y, t). Finally, the formula (2.1.49)

gives us the potential u(x, y, t) at arbitrary t,

The procedure described above gives the solution of the general initial value problem

for the KP-I equation for the class of decreasing potentials u. This procedure is not

explicit since, at the third stage, it includes the singular integral equat ions (2.1.46)

and (2.1.47). Nevertheless, it allows us to analyze the init ial value problem for the

KP-I equation with some completeness. In particular, use of t his method allows the

construction of wide classes of exact solutions of the KP-I equation (lumps, solitons,

solutions with the functional parameters, etc.). These will be considered in Section

2.3. In addition, the formulae (2.1.46) , (2.1.47) , and (2.1.49) are very useful in the

calculation of the asymptotic behavior of solutions of the KP-I equat ion as t ~ ±oo[316]. The use of such asymptotics allows the construction of act ion-angle type variables

for the KP-I equation [317].

It is noted that there is a rather simple interrelationship between the inverse

problem data f (A', A) and the scattering matrix S (A',A) introdueed in Section 1.4.

This relationship is embodied in Gelfand-Levitan-Marchenko type integral equations

[284] . Thus, if we introduce the integral operators F and Swith the kernels f(A', A)

and SeA', A), respectively, it turns out that they admit the following decompositions

into triangular factors [284]

(2.1.54)

and

58 Chapter 2

Consequently, the interrelation between S()",).) and f().',).) is expressed by a system of

linear Volterra-type integral equations [284] (seealso [318]). In particular, if f().',).) = 0,

one has S()",).) = 0 and vice versa.

The method described above allows us to solve not only the KP-I equation but

also an entire associated hierarchy of integrable equations. The hierarchy of KP-I

equations corresponds to the commutativity condition [L1, L~n)l = 0, where L1 is the

same operator as for the KP-I, namely the operator (1.3.3), while the operators L~n) are

of the form L~n) = 8t + (28x)n+... ,n = 4,5, . . .. The only difference in the integration

procedure in comparison with that for the KP-I case is that for the KP-I hierarchy, from

the equations L~n)1/J = 0, one has

8 f().', )., t) = 4.().,n _ ).n)f().' ). t)8t ~ , , ,

that is

O).t _ 0at - , (2.1.55)

f().',)., t) = e4i (A,n - An)tto:», 0), ).t(t) = ).t(O), "(;(t) = 3n().rt-1t + "(;(0)

(2.1.56)instead of (2.1.51) and (2.1.52). Substitution of (2.1.56) into equations (2.1.46), (2.1.47),

and formula (2.1.49) allows us to solve the initial value problem for the higher KP-I

equations by the same scheme as for the KP-I equation.

Note that the functions f().,).) and ).t, ).i are time independent for the whole KP­

I hierarchy of equations and infinite families of integrals of motion arise as a consequencefor all the evolution equations in the KP-hierarchy (see, for example, [317]).

In conclusion, we draw attention to a further interesting fact. In the case of 'small'

potential u(x, y, t), it follows from (2.1.31) that

Hence, by virtue of (2.1.43), for small u, one has

f().',).,t) = sgn ~).~).') 1+001+00

dxdyu(x,y,t)ei(A'-A)X-i(A'2_A2)y, (2.1.57)7l'~ -00-00

that is, the function f().',)., t) is, in fact, the Fourier transform u(N -)., ).2 - ).12, t) of

the potential u(x, y, t) . Then, for the KP-I hierarchy, formula (2.1.49) gives

'1+001+00

u(x, y, t) = .: d)' d)" ().' - ).)f().',)., 0)e i (A' - A)X- i (A'2_ A2)yH i(A,n- An)t .71' -00 -00

(2.1.58)On comparison of (2.1.58) with the solutions of the linearized higher KP-I equations ofthe form

1+001+00

u(x,y,t) = -00 - 00 dpdqu(p,q,O)eipx+iqy+irlt, (2.1.59)

Inverse Spectral Transform Method in 2 + 1 Dimensions

where O(p, q) is the linear dispersion law, we arrive at the parameterization

59

(2.1.60)

In particular, for the KP-I equation, 0 = 4{Af3 - A3) = (p3+ 3q2

). Parameterizations ofp

the type (2.1.60) play an important role in the theory of so-called degenerate dispersion

laws [319].

2.2. The Kadomtsev-Petviashvili-II equation.

Generalized analytic functions

Here, we discuss the solution of the initial value problem for the KP-II equation

wherein u{x, y, t) decreases sufficiently rapidly as x2 + y2 -+ 00 [2861 . A spectral

parameter A is introduced as in the KP-I case to yield, instead of (2.1.4),

{8y +8; + 2iA8x + u{x, y, t»X = o.

Equation (2.2.1) with u == 0 admits bounded solutions Xo of the form

(2.2.1)

(2.2.2)

where p = -(A + X), q = i(A2 - X2) , and A{A,X), B(p, q) are arbitrary bounded

functions.

Solutions X of (2.2.1), bounded for all Aand canonically normalized (X --+ 1), areA--+OO

introduced which satisfy the integral equation

X(x,y,'\) = 1- (G(·)u(·)X( ·,'\»(x,y). (2.2.3)

The formal expression for the Green function G{x, y, A) of the operator Lo = 8y +8; +2iA8x is given, in analogy to (2.1.9), by the formula

1 JJ eikx+ik'yG{x, y, A) = - (271")2 dk' dk k2+ 2kA _ ik" (2.2.4)

In contrast to the KP-I case, the integrand in (2.2.4) has poles only at the points

k = -2AR, k' = -4ARAI, and k' = k = O. Integration of (2.2.4) over k' yields

G(x.y.~) ~ -4{8(~R) [-8(Y)L. dk +8(-y) (1 dk +I:"~ dk)]

+ O{-An) [-O(Y)1-2ARdk + O{-y) ([°00 dk + [7AR dk) ] }eikX+<k2+2Ak)Y

.

(2.2.5)

60 Chapter 2

The Green function (2.2.5) depends explicitly on An = !(A + X). The correspondingintegral equation (2.2.3) likewise has such a dependence. For example, at An > 0,

X{X,yjA,X) = 1-1:001:00dx'dy'G{x-x', y-y'jA,x)U{X',y')x{X',y'jA,X) (2.2.6)

where

G{e, T/j A, X) = _-.!.. [-(J{y)10

dk+(J{-y) (rOO dk+1-2AR

dk)] eikH(k2+2Ak)l1.211' -2AR Jo-oo

Accordingly, neither the Green function (2.2.5) nor the solutions of the integral

equation (2.2.3) are analytic on the whole A complex plane. Moreover, the Greenfunction G{x, YiA, X) has no jump across the real axis AI = O. This indicates a cardinal

difference between the problems (2.2.1) and (2.1.4). Thus, one is unable to formulate aconjugation problem of the Riemann-Hilbert type for the spectral problem associated

with (2.2.1). The KP-ll equation is the first example of an integrable equation wheresuch a situation arises.

The method which, in fact, permits the formulation and solution of the inverse

problem for equation (2.2.1) was proposed by Ablowitz, Bar Yaacov, and Fokas [286] .The main idea therein is to calculate 8X/8X and then use the generalized Cauchy

formula (1.4.1O) . Following this approach, we operate on equation (2.2.3) with 1>. = ~

( 8 .8)8An + z8AI to get

8x(x y' A X) 1+00 1+008GaX ' =- -00 -00 dx'dY'ar(x-X',y-y'jA,X)U(X',Y')x(X',Y'jA,X)

1+00 1+00 8 (' r , A X)

- -00 -00 dx' dy'G(x - x', y - y'j A, X)u{x', y') x x~" .

(2.2.7)It is not difficult to show, on use of formula (2.2.5), that

8G{x, Yj A, X) _ 0"0 ipx+iqy8X - - 211'e , (2.2.8)

where 0"0 = sgn (-An), p = -2An = -(A + X), q = -4AIAn = i{A2 - X2). Note thatthe proportionality of 8G/8X to the exponent eipx+iqy is a trivial consequence of theobvious result L08G/8X = 0 and the formula (2.2.2).

On substitution of the expression for 8G/8X given by (2.2.8) into (2.2.7), we obtain

8x{x~ A, X) = F(A, X)eipx+iqy

1+00 1+00 8 (' t , A X)

- -00 -00 dx' dy'G{x - x', y - y' i A, X)u{x', y') X x:x' ,

(2.2.9)

Inverse Spectral Transform Method in 2 + 1 Dimensions

where

1+001+00

F(A).) = ;; -00 -00 dx dy e-ipx-iqyu(x,y)x(x, v.A, >.).

61

(2.2.10)

Thus, aX/a>. is the solution of an integral equation of the type (2.2.3) with free

term eipx+iqyF(A, >') instead of unity. It is emphasized that the fact that aX/a>. must

obey an integral equation of the type (2.2.9) immediately follows from the requirement

that aX/a>. satisfied equation (2.2.1). The calculation presented above gives the explicit

form of the free term in (2.2.9).

In view of (2.2:9), it is quite natural to consider functions N(x, y, A, >') which obey

the integral equation

N(X,y,A,>.) = eipx+iqy - (G(.,A,>')u( ·)N(·,A,>.))(X,y). (2.2.11)

On multiplication of (2.2.11) by F(A, >') and comparison of the resulting equation with

equation (2.2.9) (assuming that the homogeneous equation (2.2.3) has no nontrivial

solutions), we obtain

ax(x, y, A, >') = F(A >')N( A >')aA , x, y , , . (2.2.12)

The assumption about the absence of the nontrivial solutions for the homogeneous equa­

tion (2.2.3) is a very important one. For the KP-I case, the corresponding homogeneous

integral equations (2.1.14) and (2.1.15) have nontrivial solutions and the KP-I equation

possesses lump-type solutions. The existence or nonexistence of nontrivial solutions for

the homogeneous equation (2.2.3) is closely connected, as mentioned previously, with

the properties of its Fredholm determinant.

It is necessary to express N(x ,u. A, >') in terms of X(x,u, A, X). Taking into account

the symmetry property of the Green function

G(x, Yi -X - A) = G(x, Yj A, X)e-ipx-iqy,

comparison of (2.2.3) with (2.2.11) yields

N(x, Y, A, >') = X(x, u, -X, _A)eipx+iqy .

(2.2.13)

(2.2.14)

(2.2.15)

On substitution of (2.2.14) into (2.2.13), we arrive at the linear a -problem [286]

aX(x,1I.: A, X) = F(A, >')eipx+iqyX(x, v. -X,A),aA

where F(A, >') is given by (2.2.10) or, alternatively, by the formula

a 1+001+00

F(A, x) = 2; -00 · -00 dx dyu(x, y)N(x, y, ->.,-A). (2.2.16)

62 Chapter 2

(2.2.17)8'l/J(X,y,A,>.) _ F(' >.).I·(X Y ->. -A)8A - 1\ , 'I' " , •

Note that, in the terms of the function 'l/J = xei -Xx+-X2y, the a-equation (2.2.15) adopts

the form

The fact that solutions of equation (2.2.1) which are bounded in A obey the linear

a-equation (2.2.15) is most important, since it is just this a-equation which generates

the inverse problem equations for (2.2.1). Indeed, on use of the generalized Cauchy

formula (1.4.10) and appropriate substitution of the expression for 8X/8>' given by

(2.2.15), taking into account that X -+ 1 at A -+ 00, we obtain

('\) = 1 _1_ II d" d>.,F(A', >")eip'x+iq'yX(x,y, "':>",-A')

X x, Y,I\,I\ + 21l"i 1\ 1\ . A' _ A 'c

(2.2.18)

where p' = -(A' + >"), q' = i(A'2 - >.12). Then, from (2.2.1), one gets u = -2i

lim A8X/8x. From the asymptotic expansion X = 1+ A-1Xl + ... in (2.2.18), we find-X->CXl

Xl and, as a result, finally obtain

u(x, Y, t) = ~:x II dA 1\ d>' F(A, >.,t)eipx+iqyX(x, Y, ->.,-A).c

(2.2.19)

The integral equation (2.2.18) with formula (2.2.19) together solve the inverse

problem associated with (2.2.1) [2861. The function F(A, >') of two real variables AR, AIembody inverse problem data. Given F(A,>.), one can calculate X(x, y, A, >') with the

use of equation (2.2.18) and then reconstruct the potential u(x, y , t) via the formula(2.2.19).

As in the KP-I case, the inverse problem leads to a linear two-dimensional singular

integral equation. In general , the final formulae for the problem (2.1.4), (2.1.46), and

(2.1.49), and problem (2.2.1), (expressions (2.2.18) and (2.2.19», are very similar in

form. This indicates the fundamental role of linear singular integral equations in the

theory of inverse problems generated by either nonlocal Riemann-Hilbert problems or

a-problems.

The structure of the potential u(x, y, t) defined by the formula (2.2.19) is, in general,

complicated. It is a real function if the function X obeys the condition

X(x,y, ->.,-A) = X(x,y, A, >') (2.2.20)

and the inverse problem data obeys the constraint

F( -A, ->.) = F(A, >.). (2.2.21)

In order to employ the inverse problem equations (2.2.18) and (2.2.19) in the

integration of the KP-II equations, it is useful to determine the time-evolution of the

Inverse Spectral Transform Method in 2 + 1 Dimensions 63

inverse problem data. To do this, it is first noted that aX/aX obeys the same equation

(1.4.46) {with (J = 1) as X. This equation in the asymptotic region x2 + y2 ---. 00, in

view of (2.2.15), yields

Hence,

aF{~X, t) = -4i{,X3 + X3)F{'x, X,t). (2.2.22)

(2.2.23)

Thus, the initial value problem for the KP-II equation is solved by the standard

1ST scheme [286J

( 0) (2.2.10) F( \ '\ 0) (2.2.23) F( \ '\ ) (2.2.18) ,(2.2 .19) ( )u x, y, ---4 1\,1\, ---4 1\ ,1\, t ---4 u X, y, t .

The hierarchy of KP-II equations may be treated by the same scheme. For these

equations,Fn(A, X, t) = e-4i (>.n+xn)t Fn(A, X,0).

Lump-type solutions, it appears, are absent for the KP-II equation.

It is noted that, for the KP-II hierarchy, instead of (2.1.60) we have the following

parameterization of the linear dispersion laws

(2.2.24)

In general terms, the analysis of the initial value problem for the KP-II equation

has turned out to be extremely instructive and discovery of the a-method has proved a

very important step in the development of the 1ST method.

The a-formulation also reveals a deep connection between the 1ST method and

the theory of complex variables. Thus, in the case of real potentials u(x, y, t), the

a-equation, by virtue of (2.2.20), is equivalent to the following equation

ax(x,y, A, X) _ B(A X)-( A X)()A - ,x x ,y" , (2.2.25)

where B(A, X) := F(A, X) exp(ipx+iqy). But , as mentioned in Section 1.4, the solutions

of equation (2.2.25) are generalized analytic functions. The interrelation between the

a-method and generalized analytic functions theory was first pointed out by Grinevich

and S P Novikov [308, 309J.

Generalized (or pseudo) analytic functions were introduced independently by Bers

and Vekua (see the monographs [310, 311)). For our purposes, generalized analytic

functions are regarded as functions which obey an equation of the type

()/ - - -aX + A(A, A)/ + B(A,A)/ = 0, (2.2.26)

64 Chapter 2

where A and B are appropriate scalar functions . The functions contained in (2.2.26)

can be defined in various functional spaces (see [310, 311]). We note only that solutions

of equation (2.2.26) with singularities (poles, etc .) and solutions of a distribution type

are admissible.

The usual analytic functions which obey the Cauchy-Riemann condition af/aX = 0

(corresponding to A = B = 0) possess a number of well-documented properties widely

used in mathematical physics and, in particular, in the study of inverse problems. The

most important feature of generalized analytic functions is that they possess a number

of properties which are similar to the properties of the analytic functions. Accounts of

generalized analytic functions theory are presented in [310-313]. Here, we summarize

the salient properties of these generalized analytic functions which are relevant to our

purposes.

A number of properties of generalized analytic functions emerge immediately from

the observation that solutions of equation (2.2.26) can be represented in the form

to. X) = ¢(>.)eW(A,)..) , (2.2.27)

(2.2.28)

where

( \ ') = __1 JJ d>.' /\ dX' (A{>.') B{>.') j(>.'))W A, A 21l'i >.' _ >. + feN)

G

and ¢(>') is an arbitrary analytic function . Thus, the representation (2.2.27) allows

us to extend certain properties of analytic functions </>(>.) to solutions of a wide class

of equations (2.2.26). Specifically, it follows from (2.2.27) that the properties of zeros

and singular points of generalized analytic functions coincide with the corresponding

properties of analytic functions . In particular, zeros and poles of generalized analytic

functions are isolated and have finite positive multiplicities. Many boundary value

problem theorems from the theory of analytic functions may be carried over to the case

of the generalized analytic funct ions. Thus, both the maximum modulus principle and

the argument principle together with their consequences are preserved for generalized

analytic functions. The analog of the Liouville theorem concerning bounded functions

is also valid. In particular, any continuous solution (bounded in >') of equations (2.2.26)

within a certain class of funct ions adopts the form f{>' ,X) = ceW(A,)..) , where c is a

constant and w{>' ,X) is given by (2.2.28). An analog of the classic integral Cauchy

formula may also be established for generalized analytic functions .

One can introduce also a differential calculus for generalized analytic functions

[310]. This is based on the notion of the Bers derivative introduced as follows. Let us

represent the generalized analytic functions f in the form

f{>' , X) = xo{>')fo{>', X) + Xl (>')h (>', X), (2.2.29)

Inverse Spectral Transform Method in 2 + 1 Dimensions 65

where XO(A), XI(A) are real functions and fo = eWO(>"X> , h = ieW1(>"X>, where Wo

and WI are given by the formula (2.2.28) with f = fo and f = h, respectively. The

representation (2.2.29) is locally unique. The derivative j of the generalized analytic

function f at the point Ao with respect to the pair (Jo,fd is defined by

j(AO) = lim f(A) - xo(Ao)fo(A) - Xl(Ao)h(A) .>......>.0 A- Ao

(2.2.30)

This derivative j(AO) is unique for a generalized analytic function. For A = B = 0, one

has fo = 1, h = i and the Bers derivative j reduces to the usual derivative M.Analogs of Taylor and Laurent expansions and associated theorems may also be

constructed for generalized analytic functions. Indeed, the theory of generalized analytic

functions is well-developed and has a number of important applications [310-313] .

In the present context, the problem of constructing the solution of equation (2.2.26)

for given A and B is of special interest. In this connection, it is noted that (2.2.26) is

equivalent to the integral equation

f(A X) = +c (A) _1 11dA' dX'A(A')f(A', X') + B(A')J(A', x'), JO + 27l'i 1\ ).' _ A '

G

(2.2.31)

where fO(A) is an arbitrary homomorphic function. One assumption which guarantees

the unique solvability of the integral equation (2.2.31) is based on the smallness of the

norms of A and B. However, in the theory of generalized analytic functions, the unique

solvability of the integral equation (2.2.31) is proved without any such assumption about

smallness of norms of A and B. Moreover, it can be established that the general solution

of equation (2.2.6), at least, for the functions A, B of certain types , is representable inthe form

f(A ,X) = fO(A) - 2~i 11dA' 1\ dX' { XO(A') 8~' O+(A, A')+ XO(A') :A !L(A, A')} ,G

(2.2.32)

where the functions O+(A, A') and fL (A ,A') are given by the formula

and functions W±(A, A') obey certain nonlinear integral equations [31]. The functions

O±(A,A') are analogs of the Cauchy kernel to which they indeed reduce in the case

A = B = 0 (0+ -+ 1!(A' - A), fL -+ 0). For the canonically normalized function

X (X --. 1), one has XO == 1.>......00It turns out that the theory of generalized analytic functions is ideally suited to

the solution of the inverse problem for equation (2.2.1) with a real potential u(x, y, t).

66 Chapter 2

In fact, it is sufficient to use the results mentioned above in the particular case A == O.

In other words, in order to solve the inverse problem for equation (2.2.1) with a real

potential u{x, y, t), it is sufficient to study the solutions X{A) of the problem (2.2.1)

within the class of generalized analytic functions.

Using the properties of the generalized analytic functions , one can prove the unique

solvability of the inverse problem, for equation (2.2.1) with an arbitrary real potential

u{x, y, t), decreasing as x2 + y2 -+ 00. One can establish not only the smoothness of

the potential u{x, y, t) but also other key properties [309]. The theory of generalized

analytic functions gives us, in a rigorous manner, the complete solution of the inverse

problem for equation (2.2.1) and correspondingly of the initial value problem for the

KP-II equation for a class of potentials decreasing appropriately as x2 + y2 -+ 00. A

rigorous investigation of the inverse problem for (2.2.1) and of the properties of the

solutions of the KP-II equation which is not dependent upon the theory of generalized

analytic functions has been given in [320].

It is emphasized that the results presented in this and the preceding section have

been concerned with the solution of the Cauchy problem for the KP-I and KP-II

equations within the class of potentials decreasing as x2 + y2 -+ 00.

Finally, let us make a remark on the integration procedure for the KP equations

as inherited from the {I + I)-dimensional formalism. Thus linearity of the problem

(2.1.3) in 8/8y leads us to suspect that the {2+ I)-dimensional problem (2.1.3) may

be obtained from an appropriate (I + I)-dimensional matrix spectral problem via an

appropriate limiting process. This idea was realized by Caudrey in [2691 wherein it was

demonstrated that if one considers the one-dimensional N x N matrix spectral problem

1/Jxx + aD{f')1/J + v{x)1/J = 0, (2.2.33)

where v{x) = diag (Ul{X), U2{X), . . . ,UN{X», and

0 1 0 -f.-N

-1 0 1 0 0

0 -1 0 1 0 01

D{f.) = 2h 0 -1 0 1 0 0

0 0 -1 0 1

f.N 0 0 -1 0

where h > 0 and passes to the limit N -+ 00, h -+ 0 in such a way that Nh is finite,

then one obtains precisely the two-dimensional problem (2.1.3). Moreover, it was shown

that, in this limit , the inverse problem equations for the problem (2.2.33) give rise to

Inverse Spectral Transform Method in 2 + 1 Dimensions 67

the nonlocal Riemann-Hilbert problem for the case (J = i to the a-problem for the case

(J = i and to the a-problem (2.2.15) for the case (J = 1. This result clearly demonstrates

that the integration procedure for the KP equations may be inherited from a (1 + 1)­

dimensional formalism.

2.3. Exact solutions of the Kadomtsev-Petviashvili equation

Let us now consider classes of exact solutions of the KP equation which can be

calculated by the 1ST method. We start with the solutions which are given by the

formulae derived in the previous sections and which decrease as x2+ y2 - 00.

First, we consider the KP-I equation. This admits two remarkable classes of exact

solutions. The first class includes the so-called lump-type solutions and corresponds to

the case f(>..', >') = 0 so that the inverse problem equations (2.1.47) reduce to the purely

algebraic system

(x-2>'fy+12(>'f)2t+'Yf(0))xf- i i: (±xt ++ ±Xl: )=1k=l,k=i'i >'i - >'k >'i - >'k

(i = 1, ... ,n)

and the potential u(x, y, t) is given by

u(x, v, t) = 2: t(xt(x, v. t) + xl: (x, v. t)).vX k=l

(2.3.1)

(2.3.2)

The system (2.3.1) consists of 2n equations. On solution of this system for xt and xl:

and substitution of the results into (2.3.2), one obtains the explicit form of the n-lump

solutions. It may be shown that the general n-lump solution is representable in the

compact form [280]82

u(x,y,t) = 28x2lndetB,

where B is the 2n x 2n matrix with elements

BOtfJ = 8OtfJ(x - 2>.OtY + 12>'~t + 'YOt) + i(1 - 8OtfJ) >'Ot ~ >'fJ

(0:,(3= 1,. .. ,2n)

(2.3.3)

(2.3.4)

Here >'Ot = >.t (0: = 1, . . . , n), >'n+Ot = >.~ (0: = 1, . .. , n), 'YOt = 'Y.1"(0)

(0: = 1, . . . , n), 'Yn+Ot = 'Y~(O) (0: = 1, .. . , n), and 80tfJ is the Kronecker delta.

The real solutions correspond to the case >'i = >.t and 'Yi = 'Yt (i = 1, . .. , n).

A simple algebraic argument shows that the solution (2.3.3) has no singularities in the

case when all >.t are distinct.

68 Chapter 2

The simplest one-lump (n = 1) solution of the KP-I equation is of the form [280]

(2.3.5)

where x = x - 3{a2 + b2 )t - Xo, ii = Y + 6at - Yo, and a -2A1R, b = 2AlI,A1R'Y1I 'Y1I

XQ = -,- - 'Y1R, Yo = 2' .1\11 1\11

The lump solution (2.3.5) clearly has no singularities and decreases in all directions

as 1/{x2 + y2 ) moves with velocity v = (vx , vy), where V x = 3{a2 + b2) = 61A112 and

vy = -6a = I2ReAI. The existence of the solution (2.3.5) was first demonstrated by

the numerical methods in [321] .

Both the one-lump solution (2.3.5) and the general n-Iump solutions (2.3.3) cor­

respond to transparent potentials S{A', A) = 0 for the one-dimensional nonstationary

Schrodinger equation. This follows from (2.1.54).

It is readily seen that , in the case of Ao all distinct, only the diagonal elements of

the matrix B with elements given by (2.3.4) contribute to det B at t -+ ±oo. Hence, the

general solution (2.3.3) decomposes asymptotically into the superposition of n lumps,

that is,n

u(x, y, t) -----+ '"Ui(x - Vixt - XOi, Y - Viyt - Yai) ,t-+±oo~

i=1

where Ui are one-lump solutions of the form (2.3.5). Thus, the general solution given

by (2.3.3) and (2.3.4) describes the collision of n lumps. Since the asymptotics of thissolution at t -+ +00 and t -+ - 00 coincide it follows t hat the lumps do not interact.

The phase shift, which is typical in the (1 + I)-dimensional case, is absent [280]. It is

emphasized that the lumps are weakly localized bounded solutions of the KP-I equation

in contrast to the exponentially localized KdV solitons.

More general rational solutions of the KP-I equations with poles can be derived via

the formula (2.3.3). However, it is more convenient to represent them in the alternative

formn 1

u(x,y,t)=-2L( _ .( ))2'i=1x x,Y,t

(2.3.6)

where Xi{Y, t) are appropriate functions of y and t. Krichever has demonstrated in

[322] that the solutions of the KP-I equation of this type possess a remarkable property.

Thus, if the function u{x , y , t) evolves according the KP-I equation then the dynamics

of the poles in (2.3.6) is described by the Calogero-Moser system [323, 324], namely, then 2

system of n points on a line with Hamiltonian H = '" J!i... + '" { 2 )2.~2m c: x ·-x ·i=1 i#i' J

Let us now consider the second class of solutions of the KP-I equation. The simplest

solution in this class corresponds to the case !(J.L, A) = g(J.L)g(A) , where g(J.L) is an

Inverse Spectral Transform Method in 2 + 1 Dimensions 69

arbitrary function and xr == O. On multiplication of equation (2.1.46) by g{A) exp{iAx­

iA2 y + 4iA3t) and integration over A, we obtain

h~= 1+A' (2.3.7)

where

and r:h{x, y, t) = J-00 dAei>'X- i >.2Y+4i >.3 t g {A),

A{x,v, t) = _~ r: r: dA d/L 9{A)9{J,L? .ei (JL - >,)x - i ( JL2

_ >.2)Y+ 4i (JL3

_ >.3) t .

27l'Z J-oo J-oo J,L - A+ zO

By virtue of (2.1.49), in this case the solution u{x, y, t) is

( t) = .!.~ ( Ih{x,y, t)j2 )u x,y, () ,7l'8x 1+A x,y,t

(2.3.8a)

(2.3.8b)

(2.3.9)

where h and A are given by the formulae (2.3.8). The solution (2.3.9) contains the

arbitrary function g{A). It is noted that h{x, y, t) obeys the linearized KP-I equation

More general solutions of the KP-I equation which contain n arbitrary functionsn

arise out of the choice of f{J,L, A) in the form f{J,L, A) = L ge {J,L)ge (A).£=1

It is remarked that another method for the construction of solutions with arbitraryfunction dependence which relies on concepts of algebraic geometry has been proposed

in [68, 325].

The solutions of the KP equation which are of interest are not exhausted by the

solutions decreasing as x2+y2 -> 00. In spite of the absence of the solution of the general

initial value problem for the KP equation for nondecreasing but bounded u{x, y , t),

one can construct broad classes of exact solutions of this type. A procedure for the

construction of such solutions is based on the use of the nonlocal a-formalism [2711

8X{A,).) = II dA' 1\ d).'X{A' ).')R{A' ).,. A ).)8). " , , .c

(2.3.10)

The solution of the nonlocal a-problem (2.3.10) is given by the integral equation

(A ).) = 1+ _1 I·I II dA' 1\ d).' d 1\ d- X{J,L, jl)R(J,L , jl;A', ).')X '27l'i J,L J,L >.' _ A

c c(2.3.11)

70

or, equivalently, by the equation

K()..,)..') = II d)'" /\ d).'R()..', ).'j)..,).)

c

Chapter 2

+ _1 II lid /\d-d)..' /\d).,K(JL,il)R()..',).'j)..,).)2 . JL JL \f'

1l"~ JL - 1\

C C

(2.3.12)

where K := aX/a).. If now, the kernel R depends, in addition, on the variables x, y,

and t, according to the formula

(2.3.13)

then the function

or, equivalently,

( ) 2· u \aX()..,).j X, y, t)

u X, y, t = - ~ 1m 1\ ~>. .....00 uX

(2.3.14)

(2.3.15)1 a II - -u(x, u,t) = :; ax d)" /\ d)"K().. ,)..;x , y, t)x

gives a solution of the KP equation [271, 326] . This statement will be established in

Section 4.2 within the framework of the a-dressing method.

The behavior of the potential u(x, y, t) as x , y -+ 00 is defined by the properties of

the function Ro()..', ).',).., ).). In the general case, u(x, y, t) does not decrease at infinity.

In order to understand which Ro correspond to decreasing u(x, y, t) it is sufficient to

consider small u [326]. In this case, the term in (2.3.12) which contains the product K *R

is negligible and hence, one has, approximately, K()..,).) = II d>.' r.d).'R(>.',).',x, ).).c

Thus, for small u(x, y, t) , we obtain

u(x, y, t) =;: :x IIIId>./\d).d>.' /\ d).'Ro()..',).',).., ).)ei (>.I - >.).x+ ';;(>. /2_ >.2)1I+4 i (>./3_ >.3)t .

C C(2.3.16)

For u(x, y, t) as given by (2.3.16) which decrease as x2 + y2 -+ 00, it is necessary that

These conditions give 1m>.' = 1m>. = 0 for the case a = i (the KP-I equation) and

).., = -). for the case a = 1 (the KP-II equation). Thus, the solutions of the KP-I

equation which decrease as x2+ y2 -+ 00 correspond to functions Ro of the form

Ro = Ro(>., >")6(>" - ).')6(>' - ).) (2.3.17)

Inverse Spectral Transform Method in 2 + 1 Dimensions

while the decreasing solutions of the KP-II equation correspond to the case

Ro = Ro(>', X)15(A' + X),

71

(2.3.18)

where 15(A-a) is the Dirac delta-function on the complex plane. In this case, the nonlocal

a-problem reduces to the nonlocal Riemann-Hilbert problem and quasi-local a-problem,

respectively, in agreement with the results of the previous sections. It also follows from

formula (2.3.15) that the potential u(x, y, t) is real if Ro obeys the constraint

Ro(A', X, A, X) = Ro(X, A, X', A')

for the case (1 = i (the KP-I equation) and the constraint

Ro(A', Xj A,).) = Ro(-X', -A'; -X, -A)

(2.3.19)

(2.3.20)

for (1 = 1 (the KP-II equation).

Let us next consider the families of nondecreasing solutions of the KP equation

which can be explicitly calculated via (2.3.12) - (2.3.15). These solutions correspond to

factorizable kernels Ro of the form

n

Ro = L h(A', X')9k(A, X),k=l

(2.3.21)

where fk and 9k are arbitrary functions [271, 2361. In this case, by virtue of (2.3.15),

the solutions of the KP-equation are given by the formula [2711

1 8 (n )u(x,y,t) = --8 LhkXk,11' x k=l

where

hk(X,u, t) = ! ! d>' /\ dX9k(A, X)e- i>.x-';'>.2Y+4i>h

cand the functions Xk are calculated from the algebraic system

n

Xm+ L AmkXk = em,k=l

whereem (x, u, t) = JJdA /\ dXi;»; X)ei>.x+~>.2Y+4i>.3t

c

(2.3.22)

(2.3.23)

(2.3.24a)

and

Amk = 2~i ! ! dA /\ dX JJdll /\ dp, fmiA~~(Il) ei(>'-I')x+~(>.2_1'2)Y+4i(>.3_1'3)t .c c

(2.3.24b)

72 Chapter 2

The system (2.3.23) arises following multiplication of (2.3.11) bY!m{A, X)ei >.%+: >.2Y+4i>.3t

and integration over A, X.

In the general case, the solutions (2.3.22) depend on 2n arbitrary functions of

two variables. For arbitrary !k and 9k these solutions are complex. In order for the

solutions u{x, y, t) to be real it is sufficient, as follows from (2.3.19) and (2.3.20), that

gn{A, X) = !n{X,A) for the case (1 = i and In{A,X) = !n{-A, -X), gn{A, X) = 9n{-A, -X)

for the case (1 = 1. Hence, for real solutions of the KP-I equation, one has hm =

em' Consequently, the formula (2.3.22) for real solutions of the KP-I equation can be

rewritten in the equivalent, compact form [280]

where

{}2u{x, y, t) = 2{)x2 Indet A, (2.3.25)

(2.3.26)1% , , -,

Amk{x, y, t) = 8mk + -00 dx ~m{X ,y, t)~k{X ,y, t)

and ~m{X, y, t) is given by (2.3.24a). In the simplest case n = 1, one has the solution

u{x, v.t) = 2::2 In (1 + [%00 dx'I~{x', u, t)12) (2.3.27)

which generalizes the decreasing solution (2.3.9).

The soliton-type solutions of the KP-I equation correspond to the case when the

functions /k and 9k are of the form /k = IOkh(>-' + i>-'k) and 9k = 90kh(>-' - ipk), where

10k and 90k are constants [271]. The one-soliton solution corresponding to n = 1 is

given by the formula (2.3.27) with It = cb{A + ik), Imk = 0, that is, with ~(x, y, t) =cexp{kx + ik2y - 4k3t) . Another interesting solution of the KP-I equation is given by(2.3.27) with ~(x,y,t,k) = cek%+ik2Y-4k3t + J;dk'ek'%+ik'2Y-4k'3t. This describes the

annihilation of the soliton. Thus, at t -4 -00 this solution coincides with the usual

soliton, but at t -4 +00 the soliton is absent [94, 252] . This solution illustrates the

well-known instability of plane solitons for the KP-I equation [94, 248, 252, 327].

A further interesting real solution of the KP-I equation corresponds to the case

n = 2 and Ro = LZ=I Rk8{A' - J.Lk)8{A -1]k), where J.LI = -1]2 = iVl and 112 = -J.LI =iV2, Imvi = Imv2 = O. In this case [271]

{}2 ( 2 -26% )_ -6(%+vt) a eu(x, y, t) - 2{)x2 In 1+ ae cos8vy + 4(v2 _ 62) ,

where 8 = VI - V2, v = VI +V2. This solution is periodic in y and decreases as Ixl -400.

Soliton-type solutions of the KP-II equation correspond to kernels Ro of the form

Ro = L~=I Rk6{A' + iAk)8{A - illk) with Ak, Ilk > O. It may be shown on use of

Inverse Spectral Transform Method in 2 + 1 Dimensions 73

equations (2.3.22) - (2.3.24) that these solutions may be represented in the form (2.3.25)

with entries of the matrix A given by

(2.3.28)

where Cm are arbitrary constants. The one-soliton solution is given by [252]

(2.3.29)

where 0: = In \C2

• In the case A = J.L , the solution (2.3.29) is nothing but the KdVA+J.L

soliton. For A :f J.L, the soliton (2.3.29) propagates with an angle to the z-axis, The

general solution (2.3.25), (2.3.28) describes the intersection of n such "skew" solitons

and is nondecreasing along the directions x/y = Am - J.Lm (m = 1, ... ,n) as x, y ---+ 00.

The general soliton-type solutions of the KP-II equation can be represented in

another equivalent form (2.3.25), where the elements of the matrix A are given by [280]

(2.3.30)

Now, let us consider the solution of the KP-II equation which is "close" to the

soliton (2.3.29) . This solution can be calculated via the formulae (2.3.22) - (2.3.24)

with n = 1. It corresponds to the case when the functions !I(A,)..) and gl(A,)..) are of

the form [326]

!I (A,)..) = ~O(iA)b(A + )..) [ sgn (-iA - (AI - a)) - sgn (-iA - (AI + a))] ,

gl (A,)..) = ~O(-iA)b(A + )..) [ sgn (-iA - (AI - a)) - sgn (-iA - (AI + a))]

with a « AI ' Thus, the functions !I(A,)..) and gl(A,)..) are concentrated on narrow

supports near the point AI. The solution is given by [326]

(2.3.31)

where

(2.3.32)

and ¢(~) = sin~~ . This solution has rather interesting asymptotics. At 2AlY >> 8A~t,~h~ 2

A = 1+ A2

e-2>'1 (x+2>'1 ay) •y

74 Chapter 2

This corresponds to a weakly curved soliton with amplitude A~, the highest point of

which is situated on the line x = -2Alay - 2~1 In ::. In the asymptotic region u «4Alt, one can neglect the y dependence. As a result, in this region, one has

(2.3.33)

The expression (2.3.33) corresponds to the "erect" soliton with decreasing amplitude

2(Al - a)2 which propagates backwards with velocity v = \8A~a + 2: . Thus, theAl - a I\lt

solution (2.3.31) describes the propagation of a straightening wave on a curved soliton

[326]. In this straightening process, the soliton loses energy which is converted into

sound propagation. Accordingly, one can also view such waves as rarefaction shock

waves accompanied by soliton bending [326] .

It has been seen that the class of solutions of the KP equation which can be

constructed by the use of the nonlocal a-problem (formulae (2.3.11) - (2.3.15» is wide

indeed. Many of the solutions of the KP equations described above were first calculated

within the framework of the version ofthe dressing method discussed in Sections 1.2 and

1.3. For the KP equations , the kernels F and K depend on x , Z, y, and t and equation

(1.2.6a) is of the form

F(x, z, y, t) + K(x, z, y, t) +100

ds K(x, s, y, t)F(s, z, y, t) = o.

The conditions [LlO , F] = 0, [L20' F] = 0 adopt the form

8F {PF 82F

0' 8y + 8x2 - 8z2 = 0,

8F ()3F 83F

at + 4 8x3 + 4 8z3 = 0

and solutions of the KP equation are then given by the formula

8u(x, y, t) = 28x K(x, x, y, t) .

(2.3.34)

(2.3.35)

(2.3.36)

As has already been mentioned in Section 1.4, the nonlocal Riemann-Hilbert prob­

lem for the spatial contour r (real axis) is equivalent to the factorization problem

for an integral operator on a line. Similarly, one can show that equation (2.3.13) forR(A', X'; A, X) is equivalent to the system (2.3.34) [271] .

Another method for the construction of exact solutions of the KP equation and

associated equations is conceptionally close to the nonlocal a-problem procedure de­

scribed above and is known as the direct linearization method. This was first proposed

for the KdV equation [53] and is also based on the use of an integral equation. For the

Inverse Spectral Transform Method in 2 + 1 Dimensions 75

(2.3.38)

KdV equation, the integral equation is of the form (1.2.30) with an arbitrary measuredp{>..) and arbitrary contour r. This method has been generalized into the KP equationin [328].

The direct linearization method for the KP equation may be formulated as follows[55]. Let us consider the integral equation

1/J{x, v, tj k) +111/J{x ,y, tj {j(l , v))h{x, y, tj k,l, v)d{{l, v) = 1/Jo{x,u. tj k), (2.3.37)

L L

where Land d{{l, v) are arbitrary contour and measure, respectively, {j(l, v) is anarbitrary function of l, v, and 1/Jo is a solution of the system (2.1.2), that is

u1/J(}Y + 1/Joxx + Uo1/Jo = 0,

1/JOt + 41/Joxxx + 6uo1/Jox + 3{uox + a 1:00 dx'uoy{x', Y))1/Jo = 0,

where Uo{x, y, t) is some solution of the KP equation. Let the function h in (2.3.37) bedefined by

11x

h{x,y,tjk,l,v) = 2 Q dx'f{x',y,tjl,v)1/Jo{x',y,t,k) +w(y,t jk,f,a) ,

where the function w is the solution of the system

1uWy = 2[fx{a)1/Jo{a) - f{a)1/Jox{a)],

Wt = - 2[Jx;r{a )1/Jo{a ) - f;r{a)1/Jox{a) + f(a)1/Joxx{a) - 3uof{a)1/Jo(a)]

and f is the solution of the system

a fy - fxx - uof = 0,

(2.3.39)

(2.3.40)

(2.3.41)

(2.3.42)

ft + 4fxxx + 6uofx + 3 (uox - a [Xoo

dx'uoy(x', y)) f = O.

Let us assume, in addition, that the homogeneous equation (2.3.37) has no nontrivialsolutions.

It may be shown that if 1/J{x, y, t, k) is the solution of equation (2.3.37), then thefunction u(x, y, t) given by the formula

u(x, u, t) = uo(x,u, t) + ax11d{(f, v)1/J(x, y, tj {j(f , v))f(x, u,t;e. v)L L

is a solution of the KP equation [55]. This statement is proved by the straightforwardapplication to equation (2.3.37) of operators LIO and L20 of the form

£10 = uay + a; + uo(x, y, t),

76 Chapter 2

Since Ll01/10 = 0, L201/10 = 0 by the assumption (2.3.38), it follows from integration by

parts that

Lao1/1(x, y, tj k) +JJd{(l, v)Lio {t/J(x, y, tj {3(l, v))h(x, y, tj k,lv)} = 0

L L

(i=I,2) .

In view of equations (2.3.38) - (2.3.41), following some calculation, we obtain the system

of equations

£1 t/J := (u8y +~ + u(x, y, t))t/J = 0,

L2t/J := (8t + 4B; + 6u8x + 3ux + 3u [Xoo

dx'uy(x' , y))t/J = 0,(2.3.43)

where the potential u(x, y, t) is given by the formula (2.3.42). The system (2.3.43) is

compatible by construction and hence the potential u(x, y , t) is a solution of the KP

equation.

The equivalence of the formulae (2.3.37) and (2.3.42) to (2.3.11), (2.3.13) , (2.3.15)

in the simplest case Uo = 0, t/Jo = exp(ikx + ~k2y + 4ik3t ) can be proved directly.

The formula (2.3.42) allows us to construct various classes of solutions of the KP

equation corresponding to different choices of the contour L, measure df., solution Uo,

and different solutions t/Jo, W, and f ofthe systems (2.3.38) , (2.3.40), and (2.3.42). This

direct linearization method has been applied to various (2 + I)-dimensional integrable

equations in [329 - 336].

Exact solutions of the KP equation which describe the various processes of inter­

action of solitons such as resonant interaction have been derived by diverse means in

[282,337-348] . A broad class of solutions of the KP equation may also be constructed

via methods of algebraic geometry [58, 68, 325, 348-363). Here we cite only the elegant

formula82

u(x, y, t) = -28x 2 InO(vx+ vy + wt+ vo)+ const (2.3.44)

which gives solutions of the KP equation in terms of the Riemann O-function [58, 68,

325). Another method of calculation of exact solutions of integrable equations and, in

particular, of the KP equation which is essentially distinct from those considered so far

has been proposed by Marchenko [112] . This procedure is based on the construction

of solutions of scalar nonlinear equations as one-dimensional projections of solutions of

associated nonlinear integrable operator equations.

2.4. The Davey-Stewartson-I equation

Here, we apply the methods of solution described in the preceding sections to

another important nonlinear (2 + I)-dimensional system, namely, the two-dimensional

Inverse Spectral Transform Method in 2 + 1 Dimensions 77

(2.4.1a)

generalization (1.3.5) of the NLS equation. For simplicity, we shall consider the system

(1.3.5) with a = -b = 1. In this case, the system becomes

iqt+ ~(qxx + (T2qyy) + qep = 0,

irt - ~(rxx + (T2ryy) - rep = 0,

epyy - (T2epxx + (rq)xx+ (T2(rq)yy = 0,

(2.4.1b)

(2.4.1c)

where (T2 = ±1. The nonlinear system (2.4.1) is equivalent to the compatibility condition

for the following linear system [251, 252]

Ll1/7:=(8y+(T(~ ~1)8x +(~ 6))1/7=0, (2.4.2)

L2 :=(8t - i ( ~ ~1 ) 8; - ; (~ 6) 8x + ( ~: 1~))1/7 =0, (2.4.3)

(2.4.4)1

(8y ± (T8x)A± = -22' ((T8x =t= 8y)(rq).(T t

Under the reduction r = eij (e = ±1), the system (2.4.1) can be rewritten in the form

where

(2.4.5)iqt + ~(qxx + (T2 qyy) + elql2q+ q¢ = 0,

¢yy - (T2¢xx + (1 - (T2)elql~x + e(1 + (T2) Iql;y = 0,

where ¢ := ep - Iq12. Equation (2.4.5) is the long-wave limit of the Benney-Roskesequation [364] and describes the evolution of almost monochromatic, weakly higher

dimensional wave packets of small amplitude on a water surface [253]. We shall refer to

equation (2.4.1) with (T = i as the Davey-Stewartson-I (OS-I) equation and to equation

(2.4.1) with (T = 1 as the Oavey-Stewartson-II (OS-II) equation. In our subsequent

discussion, we follow mainly the papers [296, 291, 365, 366]. It will be assumed that

q(x, y , t), r(x, y, t), ep(x, y, t) - °as x2+ y2 - 00.

First, let us consider the solution of the inverse problem for the linear system (2.4.2)

in the case of the OS-I equation. In this connection, it is convenient to rewrite equation

(2.4.2) in the form

(2.4.6)( 8i

°)1/7+(°11)1/7=°'

° o, r awhere z and z are the complex coordinates z = ~ (y + ix), z = ~ (y - ix) and

8z = 8y - i8x, 8i = 8y + i8x. Equation (2.4.6) with q = r = a has a solution

1/70 = (ei~Z e-~Ai) ,where>. is an arbitrary complex parameter.

78 Chapter 2

As in the KP case, a spectral parameter must be introduced into the problem

(2.4.6). This is achieved by consideration of solutions 'l/J of the form 'l/J(z, i, A) :=

(e+i>.z 0)

X(z, z, A) 0 e-i>.i' where 'l/J obeys equation (2.4.6). The function X satisfies

the equation

LX := (~ ~ ) X - i: [(13, X] + (~ ~ ) X = 0, (2.4.7)

where (13 = (~ ~1 ) . The spectral problem (2.4.7) is the basic one in the sequel.

First of all, we note important properties of equation (2.4.7) with q = r = O. The

general solution Xo in this case which is bounded for all A is of the form

(2.4.8)

where Ol(Z), 02(Z), 03(Z), 04(Z) are arbitrary functions . Then, the operator Lo :=

(8i 0) iA

Lq=r=o = - "2[(13, '] can be represented in the factorized form [291]o o,

(2.4.9)

where D = (~ gz) and the operator E>. acts on any 2 x 2 matrix B according to

(2.4.10)

It is important in what follows that the operator E>. should be bounded for all A.

Now let us consider the solutions X(z, i, A) of the problem (2.4.7) bounded for all

A and canonically normalized (X --+ 1 at A --+ (0). Such solutions are defined by the

integral equation

X(Z,Z,A) = 1- (G(·,A)P(,)x(" A}} (z,i), (2.4.11)

where P = (~ 6) and G(z, z, A) is the Green function of the operator u: The explicit

form of the operator Gwith the kernel G(z, z, A) follows from (2.4.9) and is given by

(2.4.12)

( 8~ 1 0)The operator tr» , that is, 0 8;1 , by virtue of (1.4.10), acts according to

1 JJ ((z'- z)-l(D-

1J)(z , z) = 2rri c dz' 1\ dz' 0 o ) z' z'(i' _ i)-1 f( , ). (2.4.13)

Inverse Spectral Transform Method in 2 + 1 Dimensions 79

In view of (2.4 .10) and (2.4.13) , wemay obtain from (2.4.12) the explicit formula which

determines the act ion of the operator Gon any 2 x 2 matrix B, namely [291],

(GB)(z ,i) =

(

Bll (z', i')1 ' ,

d ' d-' z - z- z A z27l"i JJ e-i>.(z-z')-iX(.i-.i') B (z' i')

C 21 ,

i' - i '

ei>'(.i-.i')+iX(z-z') BI2(Z', i') )

z' - z

B22(Z', i')i' - i

(2.4.14)

The form of (2.4.14) manifestly demonstrates that the Green function G is bounded

for all A and has no jumps anywhere. But, importantly, the Green function G is

evidently nowhere analytic (aG/a). f 0 VA E C). To stress this property, we will write

F(z, ij A, X). By virtue of the non-analyticity of G(z, ij A, X) , the solutions Xof equation

(2.4.11) are likewise nowhere analytic on the entire complex plane. In addition, the

solutions X of equations (2.4.11) are bounded except at points at which the Fredholm

determinant ~(A) vanishes. It will be assumed that ~(A) has a finite number of

the simple zeros AI, ... ,An, that is, the homogeneous equation (2.4.11) has nontrivial

solutions at a finite number of points AI, . . . , An . At these points, the solution X has

poles. Thus, the solutions of equation (2.4.11) are of the form

_ - ~ Xi(Z, i) A _ -)

X(Z, ZjA,A) = LJ A-A' + X(Z,Z jA,A ,i=1 •

(2.4.15)

where the function Xis bounded for all A.

Let us first consider the solutions X(A) of equation (2.4.11) for values of A which

are different from AI, . . . , An. From (2.4.11) ,

aX(Z,ijA,).) (aG - -)( _ ( - aX -)(_a>. =- aX (',A,A)POX(',jA,A) z,z)- G('jA,A)POax('jA,A) z,z).

(2.4.16)

By virtue of (2.4.14),

aG = E - lrEaX x >.,

where the operator r acts accordingly to

(2.4.17)

(0, - f

O

A

I2) ,rf:= A

hi,(2.4.18)

(2.4.19)

where

j = 2~ JJdz Adif(z,i).c

It is noted that the form of the right-hand side of (2.4.17) is consistent with condition

LoaG/a). = 0 (see (2.4.8)).

80 Chapter 2

From (2.4.17) - (2.4.19), equation (2.4.16) is equivalent to

F, (~)e;"+ih) _(G(.; ~, X)p(.)8X(~;' X) ) (z,z),

(2.4.20)

where

(0 FI) _

F(A) := = -rE>.POX(·j A, A),F2 0

that is,

F ( ' ') 1 lid I\d- -i.i>'-izX ( - t) ( -, ')I 1\,1\ = 271" z ze q z, Z, X22 Z, Zj 1\,1\

C

- -i If dx d - i>.,x-i>'nY ( t) ( t·' ')- 471" ye q x, y, X22 X , y, ,1\,1\ ,

F2(A, X) = ;: II dz 1\ dzeiz>'+i.iXr(z, t, t)Xll (z, Zj A,x)

c

(2.4.21)

(2.4.22)

Equation (2.4.20) suggests the introduction of the function N(z, z; A, X) which is

the solution of the integral equation

_ ( 0 eiZ>'+iz>' ) A _

N(z, Zj A, A) = 0 \ 0_' - (GP(·)N(·, A, A))(Z,z).e-U"-U" 0

(2.4.23)

On right multiplication of (2.4.23) by the matrix (~2 ~l ), subtraction of the result­

ing equation from (2.4.20) (noting that, for the values of A under consideration, the

homogeneous equation (2.4.11) has no nontrivial solutions), we obtain

8X(Z,ZjA,X) =N( -.A ') (F2(A,X) 0 )8A z, Z, ,1\ - •o FI (A, A)

(2.4.24)

It is now necessary to determine the relation between the functions Nand X. To do

this, it is noted that the Green function G(z, z;A, X) has the symmetry [296]

where

[G(·; A, X)(f(.)E>.O)] (z, z) = [G(.jX, A)IO] (z, z)E>.(z, z), (2.4.25)

Inverse Spectral Transform Method in 2 + 1 Dimensions

On comparison of (2.4.11) with (2.4.23) and use of (2.4.25), we obtain

N(z ,z; A,X) = X(z, Zj X, A)E.x(z, z).

81

(2.4.26)

Substitution of this expression for N into equation (2.4.24) leads to the linear a-problem

[291,296]

where

aX(z, Zj..\, X) _ - _-aX = X(z,s.x,..\)F(z,z; ..\, A), (2.4.27)

F(z, Zj A,X) := E.x(z, z) (F

02 0) = ( 0 _

F1 F2(A, X)e-iz.x-iz.x

Ft (A, X~;iA+ ;" )

(2.4.28)

and the functions F1 (..\, X), F2(..\, X) are given by the formulae (2.4.21), (2.4.22).

The linear a-problem (2.4.27) is the pivotal point of our construction. However,

it is not sufficient for the complete solution of the inverse problem since equation

(2.4.27) is valid only for those values of ..\ which do not coincide with the zeros of

the Fredholm determinant. Accordingly, we must take into account the existence of

nontrivial solutions for the homogeneous equations (2.4.11). These solut ions have rather

a special structure. Indeed, let the homogeneous equation (2.4.11) have the nontrivial

solution Xi := X(z, ZjAi,Xi) at the point Ai, that is,

(2.4.29)

On use of (2.4.25), one can show that, together with (2.4.29), one has

(2.4.30)

Hence, the function Xi(Z, z)E>.; (z,z) is also a solution of the homogeneous equation

(2.4.11), but at the point Xi. For example, if equation (2.4.29) has a solution of the

form Xi = (X(i)ll 00) at the point x, then it also has a solution of the formX(i)21

at the point Xi. Further, equation (2.4.29) comprises, in fact, two independent equations

for each column of the 2 x 2 matrix Xi. Each of these equations may have its own set

of the nontrivial solutions Xi at sets of points {..\d and {jLi}' In view of this, the

representation (2.4.15) may, in fact , be written as

X(z, Zj A,X) = S(z, z, A)+ X(z,Zj A,X), (2.4.31)

82

where the matrix elements of the singular part S are given by

nl "I,. n2 "I,.

Sol = L 'I'(k)ol + L 'I'(l)~2 e-izpt-iip.t,k=l A - Ak l=l A - J.Ll

n2 "I,. nl "I,.

S _" 'I'(l)02 +" 'I'(k)ol ii'x"+iz>.,,02 - LJ LJ - e ,

l=l A- J.Ll k=l A- Ak

and ¢(k)o{J(Z,z) (a, (3 = 1 2) are appropriate functions.

Consequently, the full a-equation adopts the form [296]

aX(z, Zj A, X) (- ")F( -, ') SAaX = x Z, Zj 1\, 1\ Z, Z j 1\, 1\ + ,

Chapter 2

(2.4.32)

(2.4.33)

(2.4.34)

(2.4.35)

where the elements of the matrix S are given by (2.4.32) with substitutions of the type

(A - Ak)-l -> rr6(A - Ak).

It is also necessary to determine the relations between the function X and functions

¢(i)' The derivation is similar to that of the relations (2.1.18) and (2.1.19) for the KP

equation. First, the function Xi := X - ¢d(A - Ai) is introduced into equation (2.4.11).

We then proceed to the limits A -> Ai' As a result, in particular, [296]

(_ ¢i(Z.Z)) (-2iZ+'Yil' 0 )

lim X(z, Zj'\,'\) -~ = ¢i(Z, s) ,>'->'1 1\ - I\i 0, 2iz + 'Yi2

where the function ¢k is normalized by the condition

lim ¢k (2Z 0) = ix 2+y2

-00 0 - 2z

and 'Yi2, 'Yi2 are constants fixed by the asymptotic behavior of ¢i.

We are now able to determine the complete system of inverse problem equations.

The first such equation arises from the application of the formula (1.4.10) to (2.4.33) to

give

x(Z s.x X) = 1+ S(z z·,\) + _1_ J J d,\' t\ dX'X(z, Zj X', ,\')F(,\',x'), , , "2rri ,\, - ,\ ,

c

where S(z, z,A) is given by the formula (2.4.32) and the function F('\, X) is of the form

(2.4.28) . Passage to the limits ,\ -> Ai, ,\ -> J.Li in (2.4.35) in view of (2.4.34), produces

the residual inverse problem equations [296]

__1 JJd,\' r; dX' (X(z, Zj X', ,\')F(,\', X'nol = 62 . " , 01rrz 1\ - I\i

C

(a = 1,2),

(2.4.36)

Inverse Spectral Transform Method in 2 + 1 Dimensions 83

(

0 q) i _= -2 lim [a3, AX{Z, z; A, A)),

r 0 A--+oo

- -21 . II d>.' 1\ dX'{X{z,Zj X'~;')F{>'" X'))02 = 002 (2.4.37)7n - ILi

C

(a = 1,2).

Finally, we derive the formula for the reconstruction of the potential. From (2.4.7), one

has

and on substitution of the asymptotic expansion of the right-hand side of (2.4.35) into

this formula, we obtain, on use of (2.4.32),

"2 "1

q(x, y, t) = <L:>t>(l)12 + i L t/>(k) 11eii).k+izAkl=l k=l

+ 2~ II dA 1\ dXXll (z, Zj X, A)F1(A, X)eiiA+iz)',c

nl n2

r(x, u,t) = -i L t/>(k)21 - i L t/>(l)22e-iziil-iil'lk=l l=l

- :1r II dA 1\ dXX22(Z, z; X, >.)F2(>');)e-iZA-iiX.c

(2.4.38a)

(2.4.38b)

Equations (2.4.35) - (2.4.38) constitute the inverse problem equations for the spec­

tral problem (2.4.7). The set F = {F1(A,X, t), F2(A X, t); Ai, 'Yil (i = 1, .. . ,ndj ILk, Ak2(k = 1, ... , n2)} embodies the inverse problem data. The forward problem for (2.4.7)

consists of the calculation of F for given q(x, y, t) and r(x, y, t) via the formulae (2.4.21),

(2.4.22), (2.4.28), and (2.4.34). Given the inverse problem data F, the inverse problem

equations (2.4.35) - (2.4.37) allow the calculation of the functions X(z, z; A, X) and

t/>(k) 0 1, t/>(k)02' The formulae in (2.4.38) then give the potentials q{x,y, t) , r(x, y, t) .

The case F1 = F2 == 0 corresponds to rational q and r, that is, to lump solutions.

In that case, the inverse problem equations reduce to a purely algebraic system which

is readily solvable to produce explicit representations lump solutions. It is emphasized

that the lump solutions for the DS-I equation arise in combination with the a-equation

in contrast to the KP-I case. Investigation of the asymptotic behavior of the solution X

at z -+ ±oo also reveals that the functions F1 and F2 are the off-diagonal elements of

84 Chapter 2

the scattering matrix for (2.4.6) [291] . Hence, the lump solutions represent transparent

potentials for the two-dimensional problem (2.4.6). It also follows from the formulae

(2.4.38) that the reduction r = fij under which the general system (2.4.1) leads to

equation (2.4.5) corresponds to the following restrictions on the inverse problem data:

nl = n2 = n, (2.4.39)

<P(k)2l = e¢>(k)l2' Ilk = Xk, 'Yk2 = ikl (k = 1, . .. ,n).

Accordingly, we now have the solution of the inverse problem for the spectral

problem (2.4.7). In order to apply this to the integration of the D8-I equation, we

must, as usual, determine the time evolution of the inverse problem data F(>. ,X,t) .

This can be achieved by the same method as that adopted by KP equation. This

involves noting that aX/aX obeys the same equations as Xand then considering these

equations in the limit [z] -+ 00. There is another approach based on the use of the

formulae (2.4.38) and direct application of equation (2.4.1). For small q and r when one

can neglect the nonlinear terms in the system (2.4.1) and when Xll '" 1, X22'" 1, one

has, in the absence of lumps,

q(x, v, t) = :; JJd>'Rd>'IFl(>'R, >'I)ei>"x+iARY,

r(x, y, t) = -;JJd>'Rd>'IF2(>'R, >'I)eiARX- iAIY.

(2.4.40)

We now substitute these expressions for q and r into the linearized equations (2.4.1) to

obtain

Hence,Fl (>., X, t) = e!(A

2+ 5.

2)tr, (>.,X, 0),

F2 (>.,X, t) = e-!(A2+ 5.

2)tF2 (>.,X, 0).

On use of (2.4.34), it can also be shown that

(2.4.41)

(2.4.42)

and

a>'k = allk = 0at at

a'Yil = -2i>' .at t,

(k = 1, ... , n)

Inverse Spectral Transform Method in 2 + 1 Dimensions 85

that is,

(2.4.43)

The formulae (2.4.42) and (2.4.43) allow us to solve the initial value problem forthe OS-I equation according to the standard 1ST scheme:

I - II - III{q(x, u, 0),r(x, u, On -+{F(A, A,on -+{F(A, A, tn -+{q(x, u. t), r(x, y, tn.

The formulae (2.4.35) - (2.4.38), (2.4.42), and (2.4.43) can now be used for the con­

struction of infinite families of exact solutions to the OS-I equation. The procedure issimilar to that for KP equation.

The lump solutions of the DS-I equation (2.4.1) correspond to the case F1 (A).) =F2(A,x) =O. The inverse problem equations, as usual, are readily solved in this caseand we can obtain explicit representations for the n-lump solutions. The simplest lump

solution corresponds to n1 = n2 = 1. In this case, equat ions (2.4.36) and (2.4.37)become

(- 2iz + 1'11 )rPo1 - A rP02_ eizjj l -izl'l = 601,1 - J.l1

(0 = 1,2)(2.4.44)

(0 = 1,2)

On solution of (2.4.44), we get (rP == rP(1))

(

2iz + 1'12,1

rP= --- 0 _ 0 _

1I'(z, z) e-tzl'l-tzl'l

>'1 - PI '

(2.4.45)

where

(2.4.47)

r(x, y, t) = ( i _ ) (>. 1 _ _ 2iz - 2i>'1 t + 1'11 (0)) e-izjjl-iZl'l,11' z, z, t 1 - J.l1

eiZ(Al-JJI )+i Z(X 1-1'1)

1I'(z,i,t) = (-2iz-2iA1t+1'1l (0))(2iz+ 2iJ.l1 t +1'12 (0))+ IA - 12 . (2.4.46)1 - J.l1

Substitution of (2.4.45) into (2.4.38) now produces the one-lump solutions of the OS-I

equation (2.4.1), namely [296J,

q(x, y, t) = - i (_ 1 _ 2ii _ 2iJ.l1 t _ 1'12(0)) e iZX1 +iZA1,1I'(z, z, t) >'1 - J.l1

86

where 1l"{z, z, t) is given by the formula (2.4.46).

The factorized data

nl n2

Fl{A,X) = Lf~l){A)gil){X), F2{A,X) = Lf12){A)g~2){X)k=l l=l

Chapter 2

(2.5.1)

lead to another rich family of solutions. In particular, when the functions f~i) (A) and

f~i) (X)are superpositions of Dirac delta-functions one retrieves soliton-type solutions. It

is noted that multi-soliton solutions of the DS-I equation have been constructed within

the framework of the dressing method in [367] . Lump solutions of equation (2.4.5) with

(72 f = -1 and asymptotic behavior Iql --+ fo f 0 were first calculated via the Hirotax,y-+oo

method in [341] .

The DS-I equation and the problem (2.4.6) with nonzero asymptotics of q and r

(namely, q, r --+ 1) have been studied in [368]. A combination of the a-problemx 2+ y 2_ oo

and the conjugation problem on the unit circle IAI = 1 arises in that case.

2.5. The Davey-Stewartson-II equation

For the DS-II equation, it is convenient to represent the problem (2.4.2) in the form

(0'7 0) t/J + (

0 q) t/J - 0Ooe rO-'

where ~ = !(Y - x), 11 = !(Y+ x) .

The spectral parameter A is introduced by transition to the function X defined by

X{x,u, A) := t/J{x, y) (e-~).,e ei~'7) ' This function X obeys the equation

(0'7 0) iA (0 q)o 0e X - "2 [(73, X] + r 0 X = O. (2.5.2)

The Green function for the operator Lo = (~'7 ge) - i; [(73, '] is calculated by the

same method 'as that adopted in the previous section. Thus, in analogy to (2.4.9), the

operator Lo may be represented in the factorized form

Lo = E-;l DE)."

where D = (~'7 ge) and the operator E).. acts accordingly to

(2.5.3)

(2.5.4)

Inverse Spectral Transform Method in 2 + 1 Dimensions 87

The operator Lo as given by (2.5.3) is not bounded in >., in contrast to the operator

(2.4.9). However, this disadvantage may be set against the much simpler structure of

the operator D-l = (8f 8~1) ' As a result , the inverse operator G:= L01 is given

by the simpler formula ~(2.5.5)

(2.5.6)

that is,

A (8;IBll((,rl), 8;I(ei>.(T/-TI')BI2((,71'))).(GB)(e, 71) =

8i"1 (e-i>.(~-n B21 (e',71')), 8i"1 B22(e' ,71')

The kernel G(e,71, >') of the operator Gis the usual Green function for the operator Lo.

The main feature of the Green function (2.5.6), in comparison with (2.4.14), consists

in the absence of the X-dependence. Thus, the Green function (2.5.6) is an analytic

function on the entire complex plane of >..

Another feature of the Green function (2.5.6) is that it is non-unique. This freedom

in the definition of the Green function is connected with different possible concrete

realizations of the formal operators 8;1 and 8i"1. We can exploit this freedom in

order to construct bounded Green functions. Thus, we set 8;1 f = l~ d71'f(e, 71') and

8tf = r~ de'f(e', 71) and define the Green function G+ according toi:L d71'ei>'(T/-T/') BI 2(e, 71'))

1~ .

00 de'B22(e',71)(2.5.7)

The choice 8i"1 f = l~ de' f(f,,', 71) and 8;;1 f = J~ d71' f(f", 71') gives the Green function

G-:

J~ d71'ei>'(1/-1/') BI 2(f" , 71'))

~ .100 de'B22(e', 71)

(2.5.8)

It is readily seen that the Green function G+(x, y, >') is bounded in the upper half­

plane 1m>' > 0 while the Green function G-(x, y, >') is bounded in the lower half-plane

1m>' < O.

Now let us consider the solutions x+ and x- of the problem (2.5.2) which simulta­

neously are solutions of the integral equations

(2.5.9)

88

and

Chapter 2

x-(e, 11, A) = 1- [0-( .,A)POX-(" A))(e, 11), (2.5.10)

where 0+ and 0- are given by the formulae (2.5.7) and (2.5.8) and P = (~ 6).The solutions X+ and x- are analytic, bounded functions in the upper and lower half­

planes, respectively. Further, since G+ - G- =/: 0 at ImA = 0, X+ - X- =/: 0 at ImA = O.

Accordingly, one can introduce the function X := { ~~: ~:~ ~ ~ which is analytic on

the entire complex plane and has a jump across the real axis. It will be assumed here

that the homogeneous equations (2.5.9) and (2.5.10) have no nontrivial solutions.

We now seek the relation between the functions X+ and x- on the real axis. This

relation may be derived by the same method as that employed for the KP-I equation.

First, we set K(e,11,A) := X+(e ,11,A) - x-(e, 11, A) for ImA = O. On subtraction of

(2.5.10) from (2.5.9) and use of (2.5.7) together with (2.5.8), we obtain

K(e, 11, A) = T(e, 11, A) - [0-(-, A)P(·)K(·, A))(e,11),

where

(2.5.11)

( -1:00

d11'(Qxtl)(e, 11'),T(e, 11, A) = 1:00

de'e-iA(~-n(rxtd(e,11),

Further, let us introduce the quantity [296J

(e-iAlI 0) (0 f)

where EA(e,11) = 0 eiA~ and f(A', A) = 121 f~: ' where

2 r: r: .I .

121 (A', A) = -; L; J-00 ded11Q(e, 11, t)x2"2(e,11, A)e'A lI+'A~,

and

(2.5.12)

(2.5.13a)

(2.5.13b)

(2.5.13c)122 (A',A) =1:00

dJ.Lhl(A', J.L)!t2(J.L, A).

One can show that A obeys the same equation (2.5.11) as the jump Kce,11, A) [296] .

As a result, in the absence of nontrivial solutions of the homogeneous equation (2.5.11),

one has !:l. = K, that is,

(2.5.14)

Inverse Spectral Transform Method in 2 + 1 Dimensions 89

Thus, we have constructed a function X = {~~: i:~ ~ ~ which is analytic on the

entire complex plane A and the jump of which on the real axis is given by (2.5.14).

Accordingly, we have arrived at the standard regular nonlocal Riemann-Hilbert problem,

the solution of which is given by the formula (1.2.13). On passage in (1.2.13) to the

limit AI ~ iO, we obtain (see equation (1.2.14))

-(C -\)=1+_11+001+00

d d-\,x-(e,TI,-\')'E>d(-\',JL)'EpX <", TI, 2 . JL \ + '0 .

7l't -00 -00 JL - 1\ t(2.5.15)

The formula for the reconstruction of the potentials q, r via X- and f(-\', -\) is of the

form

(~The integral equation (2.5.15) and the reconstruction formula (2.5.16) give the

complete solution of the inverse problem for equation (2.5.2) [296]. The functions

112(-\',-\) and 121(-\',-\) constitute the inverse problem data.

The time-dependence of the inverse problem data can be found in the standard

manner. One has112(A', -\,t) = e- t <>./2+>.2)t112(-\','\' 0),

121 (-\', -\, t) = et <>./2+>.2)t121 (A', -\,0).(2.5.17)

The forward and inverse problem equations for the problem (2.5.2) and formulae

(2.5.17) give the solution of the initial value problem for the DS-II equation within theclass of decreasing solutions.

Classes of exact solutions of the DS-II equation involving functional parameters

as well as soliton-type solutions can be calculated by the standard procedure in the

case of factorized functions 112(-\',-\) and 121(-\',>'). Wider classes of solutions which

are nondecreasing at infinity can be constructed, as in the KP equation case, via

the corresponding matrix nonlocal a-problem. A particular version of this approach

coincides with the direct linearization procedure proposed for the DS-II equation in [296].

The corresponding integral equation and the solution formula are given by (2.5.15) and

(2.5.16) with the substitution II dud): ~ II dV(JL,>'), where dV(JL,A) is an arbitrary

measure on the complex plane [296]. A more general formulation is given in [55].

It has been already noted that the integral equations which arise within the non­

local Riemann-Hilbert problem method, in some special cases, are equivalent to the

Gelfand-Levitan-Marchenko type integral equations. In our case, equations (2.5.15) are

equivalent to a system of two integral equations of the Gelfand-Levitan-Marchenko type.

A mathematically rigorous solution of the inverse scattering problem for the system

90 Chapter 2

(2.5.1) was first given in [369-373] based on a method associated with the factorization

problem for integral operators.

Another approach to the solution of the inverse problem for (2.5.1) also based on

the used of the Volterra-type integral equations has been proposed by Cornille in [374].

The main feature of the corresponding inverse problem equations is that they contain

the spatial variables x and y on an equal footing in contrast to the standard Gelfand­

Levitan-Marchenko equations . In the paper [374] infinite families of exact solutions of

the DS-II equation were constructed. In particular, therein a localized solution of DS-II

in the (x, y) plane but decreasing as t ~ 00 was discovered.

The inverse problem for the system (2.5.1) and the problem of construction of

exact solutions as well as other properties of the DS equation have also been discussed

in [375-380] .

In this, as in previous sections, we have restricted ourselves to two-dimensional

matrix problems of the second order. However, one can also investigate, in the same

spirit, general N x N matrix two-dimensional problems

(By + aABx + P(x, y, t))1/1 = 0, (2.5.18)

where (12 = ±1, A is a diagonal N x N matrix with real elements, and P(x, y , t) is an

N x N matrix with elements Pij(x, y, t) which decrease as x2 + y2 ~ 00. As in the

2 x 2 case, the a-problem and the nonlocal Riemann-Hilbert problem arise in the cases

(1 = i and (1 = 1, respectively. The corresponding results have been obtained in [365,366,296].

The simplest system connected with the problem (2.5.18) is equivalent to the

compatibility condition of the linear equations (2.5.18) and

(~ + B :x + Q(x, y , t)) 1/1 = 0, (2.5.19)

where Bik = bit5ik, Qik = nik~k, and nik = (bi - bk)/(ai - ak) (i, k = 1, ... , N) . Thisis the system

where l-'ik = bi - ainik, which describes N(N - 1) resonantly interacting waves in

2+ 1 dimensions. In the case N = 3 and under various reductions, the system (2.5.20)

describes variants of three-wave resonant interaction.

The 1ST method was first applied to the system (2.5.20) by Zakharov and Shabat

in 1974 [24]. The inverse scattering problem for the system (2.5.18) in the hyperbolic

case (1 = 1 has been investigated in [381-382].

Inverse Spectral Transform Method in 2 + 1 Dimensions

2.6. The Veselov-Novikov (NVN-I) equation

91

The NVN equation (1.3.22), which is perhaps the third most important example of

an integrable equation in 2+ 1 dimensions, is markedly different in character from the

KP and DS equations. Firstly, it has a Manakov triad operator representation (1.3.21)

rather than a commutativity representation. Secondly, in order to integrate the NVN

equation, one must solve the inverse problem for the operator £1 = -8~8f/ +u(x, y, t)at a fixed eigenvalue e. For different e, we must deal with different problems since the

subspaces of solutions for the equation £11/JE = €1/JE which correspond to the different eare not gauge equivalent, in contrast to the case for the KP and DS equations.

We begin with the Veselov-Novikov (NVN-I) equation but consider a slightly more

general version than (1.3.22) with a = i, namely,

(2.6.1)

where(8- 1f) (z z):= _1 !!dZ'l\di f (z' , z' )

z '2rri z- z' ,C

(8- 1f )( s) I!!d' d-,!(z', z')i Z, Z := -2. z 1\ z , .rrt z - z

cEquation (2.6.1) may be represented in the form (1.3.21) with operator £1 = -8z8i +u(z , z, t) - €, where z = ~(x + iy), z = ~(x - iy), and operator £2 in form (1.3.23).Equation (2.6.1) is clearly formally equivalent to equation (1.3.22) with the substitutionu-€ -+ u. However, the properties of equation (2.6.1) depend in an essential manner on

the sign of e. This is connected, as will be seen, with the fact that the inverse scatteringproblem for the stationary two-dimensional Schrodinger equation

(2.6.2)

at fixed energy e is different in the cases € > 0, e < 0, and e "= O. Accordingly, these

three cases will be considered separately. We shall refer to the corresponding equations(2.6.1) as NVN-I+ (e > 0), NVN-L (e < 0), and NVN-Io (e = 0).

Before entering upon the study of equation (2.6.1), it is noted that the inverse

scattering problem for the multidimensional Schrodinger equation has been investigated

in a number of papers. Major contributions to the solution of this problem are due to

Faddeevand Newton (see, for example, [283, 383)). Important progress on the inverse

problem for the two-dimensional Schrodinger equation, that is, the problem (2.6.2) at

the fixed e, has been achieved recently in [265, 297, 384-394] both for the cases of periodic

and rapidly decreasing potentials. Combination of these results with the a-method and

the triad operator representation has allowed the integration of equation (2.6.1) by the

1ST method [265, 297, 392].

92 Chapter 2

In this section, we consider the NVN-I+ equation. The discussion follows, for the

most part, the original work in [297]. It is assumed that the potential u{x, y, t) decreases

sufficiently rapidly as x2 + y2 -+ 00.

We start, as usual, with the solution of the inverse problem for equation (2.6.2)

with fixed E = k2 > 0 [297]. Following the standard procedure, westart by introducing

a spectral parameter A into the problem by considering the solutions 'I/J of (2.6.2) of the

form 'I/J{z, i ,A) := X{z,i, t) exp[-ik{AZ + A-1 i)]. The function X obeys the equation

-8:t:8iX + ikA8iX + i;8:t:X +u{z, i , t)X = O. (2.6.3)

The solution of equation (2.6.3) with u == 0 which is bounded for all >. is of the form

(2.6.4a)

where A{>., X) is an arbitrary bounded function. Equation (2.6.3) with u == 0 also has a

solution which is defined and bounded only on the unit circle 1>'1 = 1, namely,

Xo{z, i, >., X) = II d/l /\ dJl 8{1 - 1/l12)B{>', /l)eik[(A-/l):t:+(X-p)i) ,

c(2.6.4b)

where B{>., /l) is an arbitrary bounded function.

Let us consider the solutions X of equation (2.6.3) bounded for all >. and normalized

canonically (X -+ 1 as >. -+ 00). Such solutions of equation (2.6.3) satisfy the integral

equation

X{z,s, >., X) == 1 - [G{'j >.,x)uOX{·, >., X)]{z, s), (2.6.5)

where G is the Green function of the operator Lo = -8z8i+ik>.8i+ik/>.8:t: . The formal

expression for the Green function G is

- i II ei(/l:t:+jli)G{z, i j >., >.) = 2{211")2 d/l /\ dJl /l/l - k>'Jl- (k/>')/l'

c(2.6.6)

It follows from (2.6.6) that G{z, ij >., X) is bounded in >.. Further, for 1>'1 = 1 the

integrand in (2.6.6) has an infinite number of poles. Thus , G(z, i j >.,X) has a jump

across the unit circle 1>'1 = 1. Let us denote the boundary values of G on the circle

P.I = 1 as G+ and G- given by G±{z, ij >., X) = lim G{z, i; (1± f)eit/J, (1± f)e-it/J). TheE_O

formula (2.6.6) gives, at 1>'1= 1,

G+{z, Zj >., X) - crt», ZjA,X) = 4~ II d/l r; dJl 8(1_1/l12)eik [(A- /l):t:+ (X- p) i ). (2.6.7)

c

Inverse Spectral Transform Method in 2 +1 Dimensions 93

For 1>,1 f 1, the integrand in (2.6.6) has a pole at the point J.L = k()"+1/X). As a result,

at 1)..1 f 1 one has

(2.6.8)

It should be noted that, as in the KP and DS cases, the right-hand sides of (2.6.7)

and (2.6.8) are solutions of the type (2.6.4b) and (2.6.4a) of the equation Lo¢ =0 since

Lo(G+ -G-) = 0 and LoaG/aX = 0 (1)..1 f 1, ).. f 0). Similar Green function properties

hold for various spectral problems . From (2.6.6), one can show that the Green function

G possesses the following symmetry properties

G(Z Z-·_.!. _.!.) -G-(z z-· \ ') -G(z z-· \ ')e-ik[CoX+t)z+C5.+t),ij (269), , X' ).. - " 1\, 1\ - " 1\, 1\ , • •

G(-z, -Zj -).., -X) = G(z , Zj).., X). (2.6.10)

By virtue of the above Green function properties, the solution X of equation (2.6.5)

is nowhere analytic in ).. and has a jump across the unit circle 1)..1 = 1. In addition,

the solution X of equation (2.6.5) is bounded for all ).. except at points )..i for which

the homogeneous equation (2.6.5) has the nontrivial solutions so that the Fredholm

determinant Ll()") vanishes. It is apparent that the structure of the set of the points {Adfor which the homogeneous equation (2.6.5) has nontrivial solutions is more complicated

than in the KP and DS cases. The simplest case, when this set contains a finite number

of simple points, will be considered subsequently. For the moment, it is assumed that

the homogeneous equation (2.6.5) has no nontrivial solutions.

Firstly, let us calculate aX/a), for 1>'1 =f. 1. Differentiation of (2.6.5) with respect -to

Xgives8X 8G - aX8X = a>:UX()..,)..) - GUliX'

and, on use of (2.6.8), we obtain

where

(2.6.11)

F()..,X) =

Hence,

sgn (1)..12

- 1) 1+00

dz 1\ dz u(z Z t)x(z Z ).. X)e-ik[(A+t)z+(5.+t),il.27l'i)" -00 '" , ,

(2.6.12)

ax(z, ~).., X) = eik[CoX+!)z+C5.+t),ijF().., X) _ (G(' j)..,X)u(.) 8xL~, X)) (z, z). (2.6.13)a).. 8)"

94 Chapter 2

In view of (2.6.13) it is natural to introduce a new solution N{z , z,)., X) of equation

(2.6.3) which is also the solution of the integral equation

N{z, z,)., X) = eik[(A+t)..+(X+t),i) - [G{·,)., X)u{·)N{.,)., X)]{z, z).

Combination of (2.6.13) and (2.6.14) yields

8X{z'a~).' X) = F{)., X)N{z, z,)., X).

(2.6.14)

(2.6.15)

Further, on use of the property (2.6.9) of the Green function G, we find the following

relation between the functions N and x:

N{z z ). X) = X (z z _.!. _.!.) eik[(A+t)..+(X+-t) i), , , , , X' ). .

Finally, in view of (2.6.16), we obtain the linear a-equation [297]

(2.6.16)

8X{z,z,).,X) = ( - _2. _.!.) F{\ \) ik[(>.+t) ..+(X+t),i)8), X z, z, ).'), 1\,1\ e , 1).1 =F 1 (2.6.17)

where F{)., X) is given by (2.6.13).

The relation between X+ and X- on the unit circle S (I).I = 1) can be calculated

in a manner analogous to that employed for the KP and DS equations. Thus, from

equation (2.6.5) we obtain

(2.6.18)

From the expression (2.6.7) for G+ - G-, if we set k := X+- X- , we obtain the following

integral equation

k{z, z,)., X) = ! ! dp, /\ djll5{1 - 1p,12)T{p,,)., z,z)eik[(A-I'),,+(X-M,i)

c

where

- (G-{·,)., X)u{·)k{., ).))){z, z), (2.6.19)

T{z,z,p,,).) = 2:i JJdz' /\dz'X+{z',z',).,X)u{z',z')e- ik[(A-I'),,'+(X-Mi'l. (2.6.20)

Equation (2.6.19) indicates that it is necessary to consider the function M{z, Z,)., p,) for

1).1 = 1p,1 = 1 which satisfies the integral equation

M{z,z,)., p,) = e ik[(A-I') ..+(X-Mij - [G-{·,)., X)u{·)M{·,)., p,)]{z, s), (2.6.21)

Inverse Spectral Transform Method in 2 + 1 Dimensions 95

where IAI = IJLI = 1. It is noted that the free terms in the integral equations (2.6.14)

and (2.6.21) correspond to the two types of bounded solutions (2.6.4a) and (2.6.4b) of

equation (2.6.3) with u == O.

On comparison of (2.6.21) with (2.6.19), we obtain

k(z, s, A, X) = ! ! dJL 1\ dji 15(1 - IJLI2)M(z, i, A, JL)T(z, t, JL, A) .o

(2.6.22)

Next, it is necessary to establish the connection between the functions N(z, z, A, JL) and

X(z,z, A, X). To do this, one must consider the functions if:= M exp[-ik(Az+Xz)] and

X- := X- exp [-ik ( AZ + X)] and investigate the integral equations for if and Xwhich

follow from equation (2.6.21). From the relation M(z, z,A, It) = x- (z,z,A, It) at IAI = 1,

one can show that M(z, z,A, It) = Y(It, A)x- (z,z, A, It), where Y is an appropriate linear

integral operator [297] . Accordingly, we finally obtain [297]

x+ (z, t, A, X) = x- (z,s, >.,x) +f Id>.'IR(>., >.')X-(z,s, >.', X')eik[(>'->")z+(X-X')i) ,

s(2.6.23)

1>'1 = 1

where Id>"1 := -id>" 11>"1 and R(>', >.') is an appropriate function defined at 1>'1= 1>"1 =

1.

Thus, for the problem (2.6.3), we have both the a-equation (2.6.17) and the jump

equation (2.6.23) [297] . The appearance of such a combination is a new feature inherent

in the problem (2.6.3). Recall, that for the KP and DS equations the a-problem and

conjugation problem arise separately.

Equations (2.6.17) and (2.6.23) allow us to derive the inverse problem equations

for (2.6.3). Indeed, on application of the formula (1.4.10)

x(z, z, >., X) = 1+ 2:i f ~~~ f Id>.'lx-(z,e, s: X')R(It,>")eik[(I'->")z+(il->")i]s s

x(z z -! -~)F(>" X')eik[(>"+tr )z+(X'+"fr)i)1 11 ' -, ">." >.' '+ 21l"i d>' 1\ d>' >.' _ >.

c

Whence, from (2.6.3),

u(z, z, t) = -ikoi lim (AX(z, z,>., X)).>'-00

(2.6.24)

96 Chapter 2

From the asymptotic expansion of the right-hand side of (2.6.24), we find that

u(z, " t) = :w:. [f Idplf IdAlx- (z," ~, X)R(p, ~, t)."("'-A).+('-')')]

+ JJdA A dX X(z,i, -~, -~)F(A, X,t)eik[(>.+t>z+(X+i>.iI. (2.6.25)

o

Equation (2.6.24) and the reconstruction formula (2.6.25) constitute the inverse problem

equations for (2.6.3) [297J. The functions F(A, X, t) and R(A, X', t) (IAI = IA'I = 1)

represent the inverse problem data. Note that equation (2.6.24) and formula (2.6.25)

form a complete set of inverse problem equations only in the absence of nontrivial

solutions for the homogeneous equation (2.6.5).

The inverse problem equations (2.6.24), (2.6.25) are valid both for real and complex

potentials. In the former case, the inverse problem data have additional properties [297J.

Thus, the function F(A, X) possesses the symmetry

( 1 1) 2- -F -.... - - = A F(A A)A' A ' ,

(11) - -F = - = -AAF(A A)A' A ' ,

and the function R(A,A') obeys the integral equations

R(A,A') + R(-A, -A') + f IdJ.LIR(-A,J.L)R(J.L, A') = 0,s

R(A/,A) + R(-A, - A' ) +f IdJ.LIR(-A, - J.L )R(A', -J.L) = 0,

s

(2.6.26a)

(2.6.26b)

(2.6.27a)

(2.6.27b)

where IAI = IA/I = 1.

The relation (2.6.26a) can be readily obtained from (2.6.13) ifone takes into account

the relation- 1 1

X(z,t, A, A) = X(z,s,-X' -X) (2.6.28)

which arises as a consequence of (2.6.9). Then, from (2.6.28) it follows that

x-(z,z, A, X) = X+(z, Z, A, X)+f IdJ.Llx+(z, s, J.L, ji,)R( -A, -J.L) .s

Substitution of this expression for X- into (2.6.23) produces (2.6.27a). The more

complicated calculations necessary to establish the relations (2.6.26h) and (2.6.27b)

are detailed in [297J .

Inverse Spectral Transform Method in 2 + 1 Dimensions 97

It is important to note that an inverse statement is also valid. Thus, if the arbitrary

functions F{'\, X, t) and R{'\, N, t) (I,\'I = 1,\/ = 1) obey the constraints (2.6.26) and

(2.6.27) then the function x, calculated via (2.6.24), obeys equation (2.6.5) with the real

potential u{z, z, t) given by the formula (2.6.25) [297]. These two statements completely

characterize the inverse problem data F{>',X, t) and R{>', >.',t).

The meaning of the inverse problem data becomes rather clearer if one considers

the case of a small potential u (the Born approximation). In this approximation, X'" 1and the formula (2.6.25) reduces to

u{z, t, t) = ~; f f IdJLlld>.I{JL - >')R{JL, '\)eik[(1'->,)z+(ii-);)i)

s s. 2 -~ II d>''' d,\ (X .!.) ik[(>.+t)z+();+:t)i)+ 271" >. + >. e .

c(2.6.29)

Following transition to the real variables x and y, it is evident that (2.6.29) is, in fact,

the Fourier expansion of the potential u{x, y) := JJdpdqupqeipx+iqy. For the first term,

while for the second term,

k( 1 - 1)p=- >.+=+>.+- ,2 >. >.

From these expressions, it is readily seen that the function R{>', JL) is nothing but the

Fourier harmonic u pq with Jp2+ q2 ~ 2k, while the function F{>., X) is the Fourierharmonic upq with Jp2 + q2 ~ 2k.

As in the case of the KP equation, the inverse problem data F{>',X) and R{>', >")(I,\I = 1>"1 = 1) have no obvious physical meaning. The experimentally observable

scattering characteristic is the scattering matrix . The latter is introduced in a standard

manner via the asymptotic form of the solution 1/J as T --+ 00 of the problem (2.6.2),

(2.6.30)

where r = (x, y), k = (P, q). The scattering amplitude f{k, Ikj.~·) is related to theT

inverse problem data by a system of linear integral equations [393, 394]. It follows

from these relations that the scattering amplitude f at fixed energy f = k2 depends

on the corresponding inverse data R{>., ,\') and F{'\, X) at 1,\1 = 1 [297, 394]. This has

an interesting consequence. Thus, if at a given energy f = k2 one has R(>', >") == 0

and F{>',X) is smooth and rapidly decreasing as >. --+ 0 and ,\ --+ 00 and obeys the

98 Chapter 2

constraints (2.6.26), and if, in addition, F().., X) = °at 1)..1 = 1, then the corresponding

potential (2.6.25) is a transparent one, that is, f(k, k') =°at Ik/2 = Ik'/ = e [297, 394].

Varying the function F().., X) for 1)..1 =1= 1, we obtain a wide class of potentials u(z, z)

which are reflectionless at fixed energy. The scattering associated with such potentials

is completely trivial; the phase shift is absent.

It is noted that, for real potentials u(z, z, t) , the 8-equation(2.6.17), by virtue of

(2.6.28), is equivalent to

8X(z,z,)..,X) _ F(\ X)-( - ).. X)8)" - 1\, X Z, z, , , (2.6.31)

A-I - 1where F := F()..,)..) exp[ik[()"+X)z+()..+X)z)). Hence, for real potentials u, the function

X is a generalized analytic function outside the circle 1)..1 = 1. This feature is important

in the analysis of particular solutions of the inverse problem with R().., N) == 0.

The details of the problems considered above can be found in the papers [297,

392, 393]. It should be noted that another approach to the inverse problem for the

two-dimensional Schrondinger equations has been developed in [395] .

We next consider the initial value problem for the NVN-I+ equation. The time­

dependence of the inverse problem data F().., X) and R().., N) can be found both from

the second auxiliary equation L21/J = 0, where the operator L2 is given by (1.3.23) with

0' = i, and directly by substitution of the expressions (2.6.29) into the linearized equation

(2.6.1). In the latter procedure, one should also take into account the constraints (2.6.26)

and (2.6.27) the last of which, in the case of small u(z, z, t), is of the form

R().., A) = -R(-A, -A'), R()..',)..) = -R(-).., - )..' ).

As a result, we have

F().. ,X,t) = F().. ,X,0)eik3(A3+:&+>'3+ir)t,

R(A, )..', t) = R().., s; O)eik3(A3+:&-A/3_t,:r)t.(2.6.32)

It is noted that F(A,x, t) and R().., A', t), as given by the formulae (2.6.32), obey the

constraints (2.6.26) and (2.6.37) at any time t.

In view of (2.6.32) and the inverse problem equations for (2.6.5), we are now able

to solve the initial value problem for the NVN-I+ equation by the standard procedure

(297)

u(x,y, 0)~ {F()..,X,O),R()..,A',O)} ~ {F()..,A', t),R(A,)..',t)}~ u(x,y,t) .

Inverse Spectral Transform Method in 2 + 1 Dimensions 99

Initial value problems for higher NVN-I+ equations may also be solved by the

same scheme. For such higher NVN-I+ equations , the second operator £2 is of the form

£2 = at + a;n+1 + a~n+1 + .. ., and, as a result,

F(A, X, t) = F(A, X,O)eik2n+l(>.2n+l+>.-2n-l+X2n+l+X-2n - l)t,

R(A,A',t) = R(A,A',O)eik2n+l(>.2n+l+>.- 2n-l _ >.,2n+l _ >.,- 2n- l )t

(n = 2,3, . ..).

(2.6.33)

The inverse problem equations (2.6.24), (2.6.25) and (2.6.31), (2.6.32) allow us also

to construct infinite families of exact solutions of the NVN-I+ equation which correspond

to the factorized functions F(A, X, t) and R(A,)..I, t).

Let us now discuss the discrete spectrum problem for the problem (2.6.2), that is,

the problem of nontrivial solutions of the homogeneous integral equation (2.6.5). The

general problem is still open. However, if the homogeneous equation (2.6.5) has nontriv­

ial solutions only at a finite number of simple points AI, . .. ,An, then the corresponding

potential u can be calculated explicitly, as in the KP-I and DS-I equations cases [396] .

Indeed the method of construction of such potentials is analogous to that used for the

KP and DS equations (Sections 2.1 and 2.4).

First, the function X has poles at the points AI, ... ,An, that is,

( s.x) ~Xi(Z, Z)X Z , z, A = 1 + L..- A_ A' '

i=1 •

where we assume that Ai I- 0, and IAi l I- 1, (i = 1, ... ,n). The relation

is readily established. To prove (2.6.35), we introduce the functions

II , '= (X _~) e-ik(>.z+>.-l i),.., . A - Ai '

It follows from the integral equation (2.6.5) that

[(1 + G('j A, X)u(·»JLi(' , A)] (z,z) = e-ik(>'z+>. -1 i )

1 [ • - ]- A_ Ai (1+ G(' jA, A)U('»Xi(-, A) (z, z) ,

whereG(Z s.x X) = G(z sx X) e-ik(>'Z+>. -l i), , , , , , .

(2.6.34)

(2.6.35)

(2.6.36)

100 Chapter 2

The limit A-+ Ai in (2.6.36), in view of the equation (1+C(" A, X)U('))Xi = 0, leads to

the relation

( - ) 8Xi(Z, z,Ai) . • (2637)J.Li z, Z, Ai + 8A = Xi'Yi . .

together with the orthogonality condition analogous to (2.1.19). The constants 'Yi are

fixed by the asymptotic behavior of (2.6.34) at A-+ Ai. The relation (2.6.37) evidently

gives (2.6.35).

Finally, on passage in (2.6.34) to the limit A-+ Ak and use of (2.6.35), we obtain a

system of n linear equations for the functions Xi(Z, s), namely,

(k = 1, ... ,n). (2.6.38)

On the other hand, the expression (2.6.34) together with (2.6.3) yieldsn

u(z, z, t) = -ik8..L Xi(Z, z, t).i=1

(2.6.39)

On solution of the system (2.6.38), one obtains the potential u(z, z, t) in the

compact form [3961

u(Z,z, t) = -8..8..IndetA, (2.6.40)

where the elements of the matrix A are given by

The formula (2.6.40) defines a reflectionless potential u(z, s) for arbitrary Ai and

'Yi (i = 1, .. . ,n). In order for the potential u(z, z) be a real function, the quantities Ai, 'Yk

must obey certain constraints. Thus, the poles Ai of the function X must be situated

symmetrically with respect to the origin A= 0 and be invariant under the transformation

A -+ X-1 [3961. One can show that, in this case, the potential u(z, s) given by the

formula (2.6.40) has no singularities anywhere and decreases as Izl-2 at Izi -+ 00. Such

functions (2.6.40) represent potentials for the two-dimensional Schrodinger equation

which are transparent at energy e = k2•

The formula (2.6.40) also gives rise to solutions of the NVN-I+ equation. In this

connection, on use of the equation L21/J = 0, it is seen that the Ai are time-independent

while the 'Yi have time-dependence

'Yi(t) = 3ik3 (A~ - ;t) t + 'Yi(O) , (2.6.41)

where 'Yi(O) are arbitrary constants.

Inverse Spectral Transform Method in 2 + 1 Dimensions 101

Thus, the formula (2.6.4O) for Ai and "Yi{O) , "Yi{t) which obey the constraints

described above give rational solutions which are bounded for all x and y and which

decrease in all directions. These are the lump solutions of the NVN-4 equation.

2.7. The NVN-L and NVN-Io equations

Here, we consider the other types of NVN-I equations, namely, the NVN-L (e < 0)

and NVN-Io (e = 0) equations.

For the NVN-L equation [308, 309), the linear problem is of the form (2.6.2) with

e = _r2 , r > O. The spectral parameter A is introduced by transition to the function

X{z, z, A) := 'l/JeT(>'..+A- l.i). The function X obeys the equation

( -8..8.i+ Ar8.i+ X8.. + u(z, z)) X = 0. (2.7.1)

The solutions XO of equation (2.7.1) with u == 0, bounded for all A, are of the form

( -) _ A( \ ') T[(>'-J. )..-(>.-± ).i)xo z, Z - 1\, 1\ e >. ,

where A{A,X) is an arbitrary bounded function.

Ai; in the case e > 0, we shall study the solutions X of equation (2.7.1) which are

bounded for all A, canonically normalized, and obey an integral equation of the type

(2.6.5). The formal expression for the corresponding Green function is

- i JJ ei(j.&"+il.i)G{z,Zj A, A) = 2(2 )2 dp, /\ dji. _ . _ . J.L'

7f' C pp, + ~T>.p. + ~T>:(2.7.2)

In the case under consideration, the denominator in the integrand of (2.7.2) vanishes

only at the points J.L = -ir{A - X) and p. = 0, and the condensation of zeros to the

circle IAI = 1 is absent. As a result, the Green function (2.7.2), in contrast to (2.6.6),

has no jumps. However, it is analytic nowhere in >. as in the case e > O. Indeed, the

calculation of the derivative~ (with the use of the equality F>,{ A~ AO) = 1I'0{>, - >'0))

gives

8G{z, ~j A, X) = sgn (1 -= IAI2) T[(>.-t)..-(X-t ).i)8>' 411'>' e .

Instead of the relation (2.6.9) , one has

G (z z·!. 2.) = G(z z' >. X) = G{z z· A X)e-T[(>'-f)z-(>'-±).i), 'A' >. ' , , , I , •

(2.7.3)

(2.7.4)

102 Chapter 2

By virtue of (2.7.3), the function X is also nowhere analytic. The derivative ~~ may

be calculated in the same manner as for the case e > O. Thus, differentiating the

corresponding integral equation (2.6.5) with respect to X, in view of (2.7.3), we obtain

where

8X(z~ >., X) = F(>., X)N(z, Zj >., X), (2.7.5)

F(>., X) = sgn ;~1~-1)I I dzt\ dz u(z, z)X(z, i, >., X) .e1"[eA-t)z-eX-t>zl (2.7.6)

and the function N(z, Z, >., X) is the solution of the integral equation

N(z, Z, >., X) = e1"[eA-t)z-eX-t)i] - [G(·; >., X)u(-)N(., >., X)](z, s).

On use of (2.7.4), we find the following relation between the functions N and X

N(z z >. X) = X (z Z ~ .!.) e1"[eA-t)z-e X-t)i], , , , '>.' >. .

and on substitution of (2.7.8) into (2.7.5), yields

8X(z,z,>., A) = ( - ~ .!.) F( \ ') 1"[eA-*)z-eX-t )i]8>' X z,z, A' A 1\,1\ e

(2.7.7)

(2.7.8)

(2.7.9)

where F(>., X) is given by (2.7.6). Thus, in the case f < 0, we have a pure a-problem[308,309].

The inverse problem equations are derived from (2.7.9) in a standard manner with

the use of (1.4.10) and (2.7.1). These adopt the form

- 1 II dA' t\ dA' ( 1 1) , - [(A' 1) (X' 1)-]X(z Z A >.) = 1 + - X z Z -=- - F(A A')eT-);1 z- -);1 z

, , , 21l"i ..\' - A ' '..\" N 'c

(2.7.10)and

u(z Z t) = -!.-~ JJd>. t\ dX X (z z ~ .!.) F(>' 't)eT[eA-t)z-(X-±)i) (2711)" 21l"i 8z ' 'A' A ,1\, • ••

c

Equation (2.7.10) together with the formula (2.7.11) solve the inverse problem for

the Schrodinger operator with f < 0 and, in general, the complex potential u(z,s).

Equation (2.7.10) admits a unique solution, at least, for small norm of F(A, A).

In the case of real potential u(z,z), the function X, from (2.7.4), satisfies thecondition

1 1 -X(z, s,X, X) = X(z, i, A, A), (2.7.12)

Inverse Spectral Transform Method in 2 +1 Dimensions

and the inverse data F(>., X) admit the symmetry properties

(11) 2- -F X' X = >. F(>.,>'),

( 1 1) - -F -X' -X = ->'>'F(>', >.) .

103

(2.7.13a)

(2.7.13b)

In the case of real potentials u{z, s), one can establish rigorous, stronger results for

the inverse problem associated with the SchrOdinger equation with the negative energy

e < a [308, 309] . This is due to the equivalence of the 8-equation (2.7.9), from (2.7.12),

to the equation

8x(z'a~ >., X) = B(>.,X)X(z, z, >., X), (2.7.14)

where B(A,X) = F(A, X) exp {T [(A - ±) z - (X - ±) zJ} .Equation (2.7.14) shows that the solution X(Z,z,A)) of the Schrodinger equation

with real potential u(z, s) and e < 0 is a generalized analytic function [308, 309]. This

allows us to apply the theory of generalized analytic functions [310 - 313] to the inverseproblem under consideration.

In view of the properties of the generalized analytic functions mentioned in Section

2.2, the 8-equation (2.7.14) is uniquely solvable without any assumptions on the small­

ness of the norm of F(A,X). On use of the Argument Principle (Index Theorem) for

generalized analytic functions, one can prove that, ifthe function XF(A, X) is continuous

on the complex plane A, the function A- 1F(A, X) is Lp-integrable with p > 0 inside the

unit circle, and the relations (2.7.13) hold, then the potential u{z,z) is a continuous

function and u --+ 0 as Izi --+ 00. If, in addition, the function XF{A, X) decreases as

>. --+ 0 faster than any power of >. , then the potential u(z, s) is a smooth function [308,309].

Our analysis of the KP-II and NVN-L equations shows that the theory of gen­

eralized analytic functions is very valuable in the analysis of two-dimensional inverse

problems. In all known cases where the 8-problem arises, the corresponding 8-equationis equivalent to a generalized analyticity condition. Thus, the 1ST method is closely

linked to the theory of generalized analytic functions.

In order to proceed with the integration of the NVN-L equation, one must ascertain

the time-dependence of the function F(A, X,t) . In this connection, the equat ion L27/J = 0

gives, in a standard manner ,

(2.7.15)

In view of (2.7.15), one can now solve the initial value problem for the NVN-L equation

by the usual procedure

u(x, y, 0) --+ F(A,X,0) --+ F(A, X,t) --+ u(x , y , t) .

104 Chapter 2

(2.7.16)

Here, one must use the formulae (2.7.6), (2.7.15) and (2.7.10), (2.7.11) at the appropriate

stages.

A wide class of solutions of the NVN-L equation as well as the NVN-I+ equation

can be constructed via the nonlocal a-problem. The solutions of the NVN-L equation

obtained in [396, 397] correspond to a particular case of such a construction. These are

of the formu(z, s,t) = -~OzO..lnf IdAlf(A).eT(>.Hf )-T

3(>.3+rs )t ,

s

where f(A) is an arbitrary function and S is the unit circle IAI = 1. For f(A) =C10(A - ei'f') + C20(A + ei'f'), C1C2 > 0 the formula (2.7.16) gives a potential u which

depends only on the single variable ~ = x cos cp + ysincp. In terms of the variable ~,

this potential is the usual one-dimensional soliton moving with velocity 2r2 cos3 cp along

the ~-axis. The general solution (2.7.16) can be treated as a nonlinear superposition of

one-dimensional solitons which move with different velocities [396] .

Finally, let us consider the NVN-1o equation. In fact, we can deal with this case by

the limit € --+ O. However, as one can see from the inverse problem equations constructed

above, to derive this limit is far from trivial. For this reason we choose to start with

the original linear problem, namely,

(-0%0..+u(z, z, t))1/I = O. (2.7.17)

Solutions of equation (2.7.17) with u == 0 are of the form 1/10 = Aeu z + Bei,. .. . Hence,one can introduce the spectral parameter into (2 .7.17) in two different ways. One

possibility has been considered in [399] . Therein , functions X(z,Z, A) = 1/I(z, z)e-i>.z

and x(z, z, A) = .(iJ(z, z)e- i >.% were introduced where 1/1 and .(iJ are two independent

solutions of equation (2.7.17). These functions obey the equation

(-OzO.. - iAOs + u(z, Z, t))x = O. (2.7.18)

The solutions of equation (2.7.18) with the canonical normalization can be defined

as the solutions of the integral equation

x= 1- GuX,

where the formal Green function G is given by

- i 11 ei(,.z+ji.Z)G(z, Zj A, >.) = 2(71")2 dJ1./\ djj (J1. + >')jj '

c

The integral (2.7.20) may be calculated explicitly and we get [399]

G(z, Zj A, A) = -81

.e-uZ(Ei(i>.z) + Ei( - i Az»,1r1 .

(2.7.19)

(2.7.20)

(2.7.21)

Inverse Spectral Transform Method in 2 + 1 Dimensions 105

where Ei is the so-called exponential-integral function defined by the equation 8zEi(z)

= Z-l exp z [400J.

The Green function G given by (2.7.21) is nowhere analytic although it possesses

also some useful properties. It has, however, a major disadvantage in that it is singular

at >. = O. Hence, the Green function (2.7.21) cannot be used for the construction of

bounded solutions X. It is necessary to regularize it appropriately. One method of

regularization has been proposed in [399J. With use of a regularized Green function, it

was demonstrated therein that a pure a-problem of a rather complicated form results.

The inverse problem data in this approach has the feature of increasing linearly in time

t [399] . In (398), an equation rather more general than the NVN-Io equation has been

considered.

To conclude our discussion of the NVN-I equation, let us make a few remarks

about the periodic case. The periodic problem has been studied in [384 - 390]. An

investigation of this problem was initiated by Dubrovin, Krichever, and Novikov in

the important paper [384]. Therein, the problem of the exact integration of the two­

dimensional Schrodinger equation with the Hamiltonian

(2.7.22)

was considered. The Hamiltonian (2.7.22) describes a charged particle in magnetic

and potential fields. For such a Schrodinger operator H with periodic coefficients, the

Bloch-Floke type functions '1/;, determined by the conditions

H'I/; = E'I/;,

'I/;(z +T, >.) = eikT'I/;(x,>.),(2.7.23)

where x = (Xl, X2), T = (T1,T2), and>. is a complex parameter, were first introduced

in [384].

The set of points {>'} which correspond to fixed energy E(>') = Eo form a Riemann

surface r. The inverse problem data for the problem (2.7.33) at fixed energy Eo

constitute a nonsingular Riemann surface r of genus 9 with two marked infinite points

00+ and 00_ having local parameters >.+ and >._ and a fixed set of poles with values

the residues. Given these algebraic inverse problem data, the unique operator H is

constructed such that the corresponding function 'I/;(x ,>.) obeys (2.7.23) with E = Eo

and 'I/;(x,>') = C±eA±Z±(1+ 0 (t;)) as >. --+ oo±. The reduction group Z2@Z2 which, in

the periodic case, corresponds to the inverse problem data which lead to pure potentials

and self-adjoint Schrodinger operator H = -6. + U(Xl,X2), ii = U has been found in

[390].

The combination of the above results with the L-A-B triad concept [2641 led to

the discovery of the Veselov-Novikov equation in [266] and to the detailed treatment of

106 Chapter 2

the periodic problem [266, 392]. Here , we conclude by presenting an explicit form of

solution of the NVN-I equation, namely,

u(x, y,t) = 28z8z InO(ihz +ihz +wt +~o) +C

in terms of the Prym O-function, where V1 and ih are periodic vectors and eo, C are

constants (see, for example, [392]).

2.8. The Nizhnik (NVN-II) equation

Equation (1.3.22) with (J' = 1 as set down by Nizhnik [265] was the first interesting

nonlinear integrable equation shown to be representable in the triad operator form

(1.3.21). The integration of this equation in [265] involved the solution of the inverse

scattering problem for the perturbed two-dimensional string equation ti'l/J + u'l/J =

o as treated by Fam Loy Wu in the original independent variables in [401] and in

characteristic variables in [402] . The solution of this inverse scattering problem on the

half-line had been given earlier in [403] (see also [371]) . In these papers, the solution of

the inverse scattering problem was based on the use of transformation operators. In this

approach, the inverse problem formulation consists of a system of linear,Volterra-type

integral equations. Subject to certain assumptions on the potential u, one can establish

the unique solvability of these equations. The solution of the inverse scattering problem

for the perturbed string equation with a 2 x 2 matrix potential was obtained by a similar

procedure in [404].

Here, we consider the NVN-II equation

(2.8.1)

within the framework of.the nonlocal Riemann-Hilbert problem and a-equation method

[405] . In (2.8.1), (8i1f)(e, 1]) :=J~oo de'!(e', 1]) and (8;;1!He, 1]) :=J:!.oo d1]'!(e, 1]').

The linear problem which corresponds to the NVN-II equation (2.8.1) is of the form

-8~8T/'l/J +u(e,1], t)'l/J = 0,

where 8~ = 8x - 8y , 8T/ = 8x + 8y , ~ = !(x + y), 1]+ !(x - y). The spectral parameter

A can be introduced in two ways, namely, by transition to the function X(e,1], A) :=

'l/J(e, 1])e-i>.~ or to the function J.l(e, 1],A) := 'l/J(~,1])e-i)\T/. In the first case,

(-8~8'7 - iA8T/ + u(e,1], t))X = 0

while, for the function u; one has the equation

(-8~8T/ - iA8~ + u(e,1], t))J.l = o.

(2.8.2)

(2.8.3)

Inverse Spectral Transform Method in 2 + 1 Dimensions 107

In what follows we will use both equation (2.8.2) and equation (2.8.3).

Let us consider the solution X of equation (2.8.2) which is bounded for all >. and

canonically normalized. Such a function satisfies the integral equation

x(~, 71, >') = 1 - (G(·,>')u(')X(" >'))(~, 71),

where the Green function G is given by the formula

1 JJ eik'~+ik'7G(~, 71, >') = (271-)2 dkdk' kk' + >.k ·

(2.8.4)

(2.8.5)

In analogy to (2.8.4), equation (2.8.3) admits a solution 1L defined by the integral

equation

(2.8.6)

where the Green function 6(~, 71, >') differs from the Green function (2.8.5) only by the

change ~ +-+ 1/ so that 6(~, 71, >') = G(1/,~, >.).

It follows from (2.8.5) that the Green functions G(~, 71, >') and 6(~, 71, >') have jumps

across the real axis 1m>' = O. One also concludes from (2.8.5) that it is possible to

construct the Green functions G+, 6+ and G- ,6- bounded and analytic in the upper

and lower half-planes, respectively. These are of the form

G+(~,71,>') = (}(-~)(}(71)e-i>'~ , (2.8.7a)

G- (~, 71, >') = -(}(~)(}(71)e-i>'~ , (2.8.7b)

and

6+ (e,1/,>') = (J(e)8( _1/)e- i A'7 , (2.8.8a)

6-(~, 71, >') = -(}(~)(}(71)e-i>''7. (2.8.8b)

With the aid of these Green functions, we define solutions x+, x-, x+, x- of

equations (2.8.2) and (2.8.3) which are also solutions of the integral equations

(2.8.9)

and

(2.8.10)

By virtue of the boundedness and analyticity of the Green functions G± and 6±,the solutions x+(~, 1/, A), 1L+(~, 1/,>') and X- (~ , 71, A), 1L-(~, 71, >') are also bounded and

analytic in the upper and lower half-planes 1m>' > 0 and Imx < 0, respectively. It

will be assumed that the homogeneous equations (2.8.9) and (2.8.10) have no nontrivial

108 Chapter 2

{X+, ImA < 0

solutions. Accordingly, we can introduce the function X{~, TJ, A) :=X-, ImA < 0

which is analytic and bounded on the whole complex plane (ImA :f 0) and has a

jump across the real axis ImA = O. Analogously, we can introduce 1£ according to

{1£+, ImA > 0

1£ := . By this means, we arrive at the standard regular Riemann-1£-, ImA < 0

Hilbert problem [405].

Let us consider the function X. In order to reveal the nature of the Riemann-Hilbert

problem (local or nonlocal), one must calculate the quantity K{~, TI, A) := X+{~, TI, A) ­

X- (e, TI, A) for real A. To do this, we employ equations (2.8.9) and use the equality

(ImA = 0) (2.8.11)

which follows from (2.8.7). Consequently, we have

K{e ,TJ,A) =- JJd1.'dTJ'(){TJ-TJ')ei>.(~-e)u{~',TJ')X+{~',TJ',A)

- (G-{·, A)u{·)K{·,A»{~, TJ) .

(2.8.12)

Next, we transform the first term in the right hand side of (2.8.12) using the well-known

representation of the step function

We obtain

i 1+00 ei >" (l'/- r1')()(TJ - TJ') = - 211" -00 dA' >..' _ iO . (2.8.13)

where

(2.8.14)

(2.8.15)

Now, we need the function 1£. It is noted that the function p- = ei>'l'/IL-{~ , TJ, A)

obeys the equation

(2.8.16)

where Gij(e, TJ) := G-{~, TJ, A = 0) = -(){~)(){TJ). Comparison of (2.8.16) with (2.8.14),

on the assumption that the homogeneous equation (2.8.16) has no nontrivial solutions,

yields

(2.8.17) .

Inverse Spectral Transform Method in 2 + 1 Dimensions 109

We must next establish the relationship between t he functions X- and J.t-. To

this end, we calculate, the jump J.t+ - J.t- across the real axis. Repeating the above

computations and using the equality G+(~,'I7,A) - G-(~,'I7,A) = O(~)e-iA'1, together

with the analog of equation (2.8.16) for X:= xeiA~, we obtain

where

(2.8.18)

(2.8.19)

It is noted that the jump of the function X is expressed via J.t- and the jump of the

function J.t is expressed through X- .

The relation (2.8.18) defines the nonlocal Riemann-Hilbert problem for the function

{J.t+, ImA > 0 hi h . I' id hi ' Th I' f hiJ.t = W lC IS ana ytic outsi e t e rea axis. e so ution 0 t ISJ.t-, ImA < 0

Riemann-Hilbert problem is given by the formula (1.2.13). By projection onto the real

axis (1m>. -+ -0), we obtain the formula (1.2.14), that is, the relation

between the functions J.t- and X- . On substitution of (2.8.20) into (2.8.17), we find

(2.8.21)

with

(2.8.22)

and

R(t 'YI A' A) = _l_j+<Xl j+<Xl dpdp' p. (A' p')F (p' A)ei(A'-A)Hi(p' -p)'1""Il' 2 ' ,J '0 2, l, ,7l'~ -eoc -<Xl P - p - ~

(2.8.23)

where the functions F l and F2 are given by the formulae (2.8.15) and (2.8.19).

Thus, for the function X, we have a nonlocal Riemann-Hilbert problem with jump

across the real axis defined by (2.8.21). The formula (1.2.13) gives the solution to this

problem. By projection onto the real axis, we obtain, finally,

- 1 j+<Xl , T(~, '17, A')X (~, '17, A) = 1 + 27l'i -e-oo d); >.' _ A+ iO

+ _1_j+<Xlj+<Xl dA'dA"R(~, '17 , A", )..')x-«(., 'I7 , )..11) , (2.8 .24)27l'i -<Xl -<Xl >.' - A+ iO

110 Chapter 2

where the functions T(~, 'TI, A) and R(~, 'TI, A',A) are determined by the formulae (2.8.22)

and (2.8.23). The reconstruction formula for the potential u(~, 'TI, t) can be derived from

the equality u(~, 'TI) = i8'1 lim AX-(~, 'TI, A) which follows directly from (2.8.2) and the>......00

asymptotic expansion of the right hand side of (2.8.24). It is of the form

u(~, 'TI, t) = - :'TI [i:oo

dAT(~, 'TI, A) +i:ooi:oo

dAdA'R(~, TI, A',A, t)X-(~, 'TI, AI)] .

(2.8.25)

Equation (2.8.24) and the reconstruction formula (2.8.25) comprise the inverse

problem equations for (2.8.2) [405J. Similar inverse problem equations arise if one starts

with the functions J.L which obeys equations (2.8.3). Thus, the two ways of introducing

the spectral parameter (equations (2.8.2) and (2.8.3)) lead to equivalent results .

The time-dependence of the inverse problem data is found in a standard manner

from the second auxiliary linear problem L21/J = 0, where the operator L2 is given by

(1.3.23) with (J' = 1. One has

F}(A', A, t) = F}(A',A, 0)e- i (k t >.3_ k2>,'3)t ,

F2(A/, A, t) = F2(A/, A, 0)e- i (k2>.3_ k t >.'3)t .

(2.8.26)

The formulae (2.8.26) and the inverse problem equations (2.8.24) and (2.8.25) allow

us to solve the Cauchy problem for the NVN-II equation (2.8.1) via the standard 1ST

scheme.

It is emphasized that the constants k} and k2 in the NVN-II equation (2.8.1) are

arbitrary. In the particular case k} = k2, one retrieves the symmetric version of the

NVN-II equation, while in the case k2 = 0, we have the simpler equation [405J

(2.8.27)

A generalization of the NVN-II equation which corresponds to a slightly modified

operator L2 has been considered in [405J. The results presented above are also valid in

that case with only a minor modification of the time-dependence of the functions F}

and F2 [405J .

Exact solutions of the NVN-II equation can be now found by the usual procedures.

One of the simplest solutions of the symmetric NVN-II equation (kl = k2 = 1) is of the

form [265]

u{~, TI , t) = 28~ In(a +{3)8'1 ln(a + {3),

where a(~, t) and {3(TI, t) are arbitrary solutions of the equations

(2.8.28)

{3t - {3'1'1'1 = O.

Inverse Spectral Transform Method in 2 + 1 Dimensions 111

Results analogous to those described above can be also obtained for the generaliza­

tions of the NVN-II equations similar to the NVN-I± equations . The inverse problem

for the corresponding linear equation

{-8e8" + u{~, 71))V; = ev;

is reduced to a nonlocal Riemann-Hilbert problem.

Finally, we remark that one can study in a similar manner the problem

(2.8.29)

(2.8.30)

both in the case of decreasing potentials Rand Q and in the case R --> 0, Q --> Qoo i= 0

as e+ 712 --> 00 [406].

Chapter 3Other Integrable Equations and Methods of Solution

in 2+1 Dimensions

3.1. The multidimensional resonantly-interacting three-wave model

Models which describe the resonant interaction of wave packets are of great interest

in physics. Such resonant interaction arises, for instance , in the case of three waves if

the conditions

where ki are the wave vectors and W i are the frequencies of the waves, are simultaneously

satisfied.

The three resonantly-interacting wave model which is described by the system of

equations8ql - ~ - - 0at + Vl ql + 'Ylq2q3 = ,

8q2 + -" + - - 0fit V2 v q2 'Y2ql q3 = , (3.1.1)

8q3 - ~ - - 0at + V3 q3 + 'Y3QlQ2 =

is one of the simplest forms of nonlinear integrable systems. On the other hand , it is

one of the richest systems both from the point of view of the variety of the processes it

describes and of the structure of its solution manifold.

In the one-dimensional case, this model (the system (1.1.11)) has been studied in

detail in [11, 22, 125, 126]. In particular, exact solutions which describe the interaction

and decay of such waves as well as more complicated features have been constructed.

The multidimensional system (3.1.1), however, possesses a highly complex structure.

Zakharov and Shabat first demonstrated the applicability of the 1ST method to the

system (3.1.1) in [24]. Zakharov [254] then formulated the inverse problem equations

for the case d = 3 and found solutions with functional parameters of the system (3.1.1).

Such solutions were subsequently also obtained in [407]. Alternative auxiliary linear

113

114 Chapter 3

formulations for the two-dimensional (d = 2) system (3.1.1) which are equivalent to

those used in [24] but which contain the spatial variables in a more symmetric manner

were proposed by Ablowitz and Haberman in [251] .

A basis for the complete solution of the three-dimensional (d = 3) system (3.1.1)

has been set down by Cornille [255, 408 - 410]. He proposed a new pair of auxiliary linear

operators for (3.1.1) which contains the independent variables XI, X2, X3, and X4 == t on

an equal footing. These operators are [255]

c0

0) CPI2

~.).L+ = ~ £2 o + 0 0

0 £3 P31 0

C0

~) + (~,0 PI')

L_ = ~ £3 0 o ,0 £1 0 P32 0

(3.1.2a)

(3.1.2b)

where £i = L:~=I ai", IJ~'" and a i", are real constants. The equality [L+ , L_] = 0 is

equivalent to the system

£jPj,j_1 + Pj,j+IPj+I,j-1 = 0,(3.1.3)

£jPj_l ,j +Pj-I,j+IPj+I,j = 0,

where the indices are cyclic (j = 1,2, 3jj - 1 = 3,1, 2j etc) . In general, the system(3.1.3) describes six resonantly interacting waves Pij . However, it is not difficult to

verify that, under the reduction

PI3 = 'Y3Q2, P21 = 'Yl Q3, P32 = 'Y2QI,

and

£I = at + vV, £2 = at + V3 V, £3 = at + VI V,the system (3.1.3) reduces to the system (3.1.1) with d = 3. The operators L+ and L_

given by (3.1.2) are equivalent to those proposed in [24] . However, the operators (3.1.2)

turn out to be much more convenient for the analysis of the inverse problem.

The main achievement in Cornille's work was the derivation of the inverse problem

equations for the system

(3.1.4)

with operators L+ and L_ of the form (3.1.2). In particular, Cornille considered the

case of degenerate kernels and constructed the corresponding exact solutions of (3.1.1)

which decrease in all directions in three-dimensional space [255].

Other Integrable Equations and Methods in 2+1 Dimensions 115

The complete analysis of the three-dimensional system (3.1.1) was performed by

Kaup in a series of papers [256, 411-413]. Therein, it was noted that Cornille's construc­

tions are radically simplified if one introduces the characteristic coordinates Xi defined

by

o --0 := Ot + Vi'i/,Xi

The system (3.1.4) then becomes

(i=I,2,3) . (3.1.5)

Ok'i/Ji + 'Ykiij'I/Jk = 0,

while the system (3.1.1) adopts the form

(3.1.6)

(3.1.7)

where Oi == OjOXi and all indices i,i, k are cyclic (i,i,k = 1,2,3).

In characteristic coordinates, the dressing transformation analogous to that pro­

posed in [255] is

(3.1.8)

where the index n is associated with the n-th independent solution, u{n) is an arbitrary

solution of the system (3.1.6) with qi == ° (i = 1,2,3), and X := (Xl, X2,X3). The

solution of the inverse problem is given by the integral equations

k(n){ -) F{n){ )i S, X + i Xi, Xn + S

(n, i = 1,2,3) (3.1.9)

where Fi{n) (Xi, Xn) are arbitrary functions. The potentials qi are reconstructed by the

formulae''''fiQj{X) = kii){O,X),

'Ykiij{x) = k~k)(O,X),(3.1.10)

where the indices i, i. k = 1,2,3 are cyclic.

Characteristic coordinates are also very convenient for the calculation of exact solu­

tions of the system (3.1.7). In particular, the solutions which correspond to degenerate

kernels F of the type

F~m)(u,v) = {-'Ym9m{V)gn(u), m =1= n

0, m=n,(3.1.11)

116 Chapter 3

(3.1.12)

(3.2.1)

where 9n(U) are arbitrary functions, have an elegant explicit form. Thus, equation

(3.1.9) and formulae (3.1.10) give [255, 411]

_ 1+1'jGj(Xj)qj(X) = 9i(Xi)9k(Xk) D(Xi, Xj, Xk)'

where Gj(Xj) := J;; dsgj (S)9j (S) and

It may be shown that, under the additional constraint Gj(v) < 1 ('v'v) , one has

D > 0, and hence, the solution qj given by (3.1.12) is nonsingular. The solutions of the

system (3.1.1) of the type (3.1.12) were first constructed in [254].

Characteristic coordinates have also played a significant role in the study of the

initial value problem for the system (3.1.7). The complete solution of the inverse

scattering problem for (3.1.6) and of the initial value problem for the nonlinear system

(3.1.7) have been given by Kaup in [256]. However, the forms of solution of these

problems are very involved and, accordingly, are not reproduced here. Rather, the

interested reader is referred directly to the relevant papers [256, 412]. It is noted that

lump-type solutions of the system (3.1.7) different from (3.1.12) have been considered

in [414, 415] .

3.2. The Ishimori equation. The Hirota method

A model which generalizes the one-dimensional continuous isotropic Heisenberg

equations (1.1.27) to two spatial variables x, y and which is also integrable by the

1ST method was discovered by Ishimori in [257]. This model is described by equation

(1.3.12), that is,- - - 2- - -S, + S x (Sxx + a Syy) + 'PySx+ 'PxSy = 0,

2 2 - - -'Pxx - a 'Pyy + 2a S(Sx x Sy) = 0,

where 55 = 1 and a2 = ±1. Equation (3.2.1) is equivalent to the compatibility

condition for the linear system [257]

(3.2.2a)

(3.2.2b)

where P(x, y, t) := (jS(x, y, t).

The Ishimori equation (3.2.1) is of considerable interest since it represents the first

example of an integrable spin system in the plane. One interesting feature is that

it possesses topologically nontrivial solutions. Thus, let us consider the solutions of

Other Integrable Equations and Methods in 2+1 Dimensions 117

equation (3.2.1) which tend to So = (0,0, -1) as x2 + y2 --+ 00. All infinitely far points

of the plane are identified for such solutions, and the (x, y) plane is equivalent to the

two-dimensional sphere 82. On the other hand, the real functions 8 1, 82, 83, by virtue

of the constraint 8~+8~ +8§ = 1, also take their values on the two-dimensional sphere

8 2 . As a result, for the class of boundary conditions under consideration, the spin

vector S(x,y) defines a mapping B2 --+ B2. All these mappings can be separated into

classes of topologically nonequivalent mappings. The invariant characterization of such

separation is given by the values of the topological charge (topological index, mapping

degree) (see, for example, [416))

1 II -- -Q = 471" dxdy8.(8z x 8y ) . (3.2.3)

The quantity Q is integer-valued and is, associated with the covering number of the

sphere 8 2 by the spin-vector S(x, y) for coordinates x, y running through all points on

the plane (x, y).

Thus, the solutions of equation (3.2.1) are characterized by the topological charge

Q given by (3.2.3). The solutions with Q = °are topologically trivial, while the

topologically nontrivial solutions with Q = ±1,±2, ... cannot be transformed into

trivial solutions by a small deformation. It is readily verified that aQ/at = 0, that

is the topological charge (3.2.3) is an integral of motion for the model (3.2.1). Thus , the

topological properties of the solutions of equations (3.2.1) are conserved in time. It is

noted that topological invariants are very important in certain theoretical models (see,

for example [416)).

Another interesting property of the system (3.2.1) is its gauge equivalence to the

OS equation [329, 417]. Specifically, the system (3.2.1) with S(x, y, t) = O'2S(x ,y, t/2)

is gauge equivalent to equation (2.4.1). Indeed, one can verify that equation (3.2.2a)

with S(x,y,t) = O'2§ (x , y,t / 2), that is, the equation

is converted into equation (2.4.2) via the gauge transformation .,j; --+ g(x, y, t)'l/J, so that

L(2.4.2) = g-l L1g. The relationship between the coefficients of the operators L1 and L 1(2.4.2) is given by the formulae

and

p (x, v, t/2) = -gO'3g-1

(0 q(x,u,t)) -1 -1

= O'3O'g gz - 9 gy.r(x, y, t) 0

Furthermore, equation (3.2.2b) with t --+ t/2, P = 0'2 P(x, y,t/2)

equation (2.4.3) under this gauge transformation.

(3.2.4)

is converted into

118 Chapter 3

(3.2.5)

(3.2.6)

Hence, the pair of equations (2.2.2) with P = a2 P{x, y, t/2) is gauge equivalent to

the pair (2.4.2t, (2.4.3). This establishes the gauge equivalence of equation (3.2.1) with

S{x,y,t) = a2S (x , y , t / 2) to equation (2.4.1).

The gauge equivalence of the (2 + I)-dimensional equations (3.2.1) and (2.4.1)

generalizes the gauge equivalence of their {I + I)-dimensional versions [36] . As in the

one-dimensional case, the notion of gauge equivalence plays a key role in the theory of

(2 + I)-dimensional integrable nonlinear equations.

The Ishimori equation (3.2.1) can be investigated by the methods described in

the preceding chapter. Like the DS equation, equation (3.2.1) has markedly different

properties for a2 = -1 and a2 = 1. We shall refer to (3.2.1) in these cases as the Ish-I

(a = i) and Ish-II (a = 1) equations, respectively.

For the Ish-I equation, the auxiliary linear problem (3.2.2a) can be written in the

form

(8i 0) 1t/J + "2 {P(x, y) + (3){8z - 8i )t/J = 0,o 8z

where z = ~(y + ix) and z = ~(y - ix). The spectral parameter A may be introduced

(e-

i AZ 0)into the problem by transition to the function X{z,Z,A) := t/J{z, z) .\ _ . The

o e''''Zresulting equation for the function X is of the form

(8

i 0) X _ iA [a3, X]- (P(z ,Z, t) + (3)(iA +8i - 8zh = o.o e, 2

This equation is very similar in form to the corresponding equation (2.4.7) for the DS-I

equation. In particular, the terms which are independent of the potentials coincide.

Hence, the Green functions in the corresponding integral equations of the type (2.4.11)

likewise coincide . As a result, the analysis of the Ish-I equation is, in many respects,

analogous to the corresponding analysis for the DS-I equation as given in Section 2.4.

In particular, for the Ish-I equation, a pure a-problem may be constructed which

generates the inverse problem equations [417] . The Ish-II and DS-II equations are

likewise analogous.

It is convenient at this stage to introduce an ingenious method of calculation of

exact solutions of nonlinear integrable equations, namely, the celebrated Hirota method.

This technique was first introduced for the construction of the N -soliton solutions of the

KdV equation in [19]. It has been subsequently applied to a wide diversity of nonlinear

integrable equations (see the reviews [48, 49] and [105, 107]).

The Hirota method involves the transformation of the original equation into a

bilinear form amenable to series solution. For the KP equation, the appropriate change

of dependent variable is [282] (cf. (2.3.25))

u{x, y , t) = 2(ln f) ",,,,. (3.2.7)

Other Integrable Equations and Methods in 2+1 Dimensions

In the terms of the variable I the KP equation becomes

119

(3.2.8)

(3.2.9)

The bilinear form (3.2.8) can be rendered more compact if one introduces operators

D:z; , Dy , and D, which act according to

D~D;:D~a(x, y, t) .b(x, y, t)

:= (8x - 8~)n(8y - 8y ' )m(8t - 8t, )ka(x, y, t)b(x', y', t')1 x' = x .y' = yt' = t

With the use of such operators, equation (3.2.8) may be rewritten in the form [282]

(3.2.10)

The two most important features of the form (3.2.10) of the KP equation are its

bilinearity in the dependent variable I and the hidden bilocality in the independent

variables x, y, and t. The introduction of the operators Dx , Dy, and D, not only allows

us to rewrite (3.2.8) in the compact form (3.2.10), but it suggests a deep intrinsic

relationship between the nonlinear KP equation and the simple bilinear equation in the

extended six-dimensional space (x, y, t; x' , y', t').

The bilinear form (3.2.10) is very convenient for the calculation of exact solutions.

These can be found in the series form I = 1 + fit + f2 h + .... It is readily seen that

It = I:~=1 er/i, where TJi = ki(x+PiY- (k~ + 3a2pn t ) and ki,Pi are arbitrary constants.

The function I constructed by such an iteration scheme will be the solution of equation

(3.2.10) if the series which defines I terminates [48, 49]. The various algebraic properties

of the operators D x , D y , and D, are very useful in the construction of exact solutions

by the Hirota method.

The N -soliton solutions of the KP equation have been constructed by the Hirota

method in [282]. For these solutions, (d. (2.3.23), (2.3.30))

N

I = I: exp [I: /liTJi + I: /li/ljAij] ,;;=0 i=l l:5i:5j

where TJi = ki{x + PiY - (k~ + 3a2pnt) and

A . . (ki - kj)2 - a2(pi _ pj)2e ., - -;:---:"7~---;;-7--":""::";:-;;-

- (ki + kj)2 - a2(pi _ pj)2'

(3.2.11)

Now let us consider the Ishimori equation (3.2.1) . The appropriate bilinearization

is given by the formulae [257]

S 'S 21g1+ t 2= 11 -,+gg

S _11 - 993 - II + 99'

(3.2.12a)

120

C{)x = -2i0-2Dyq.f + g.g) ,ff+gg

_ 2· Dx(f.f + g.g)C{)y - - t f f + gg .

In the terms of these variables, equation (3.2.1) adopts the form

(iDt + D; + 0-2D;)(f.f - g.g) = 0,

. 2 2 2 -(tDt + Dx + 0- Dy)f.g = 0

Chapter 3

(3.2.12b)

(3.2 .13a)

(3.2.13b)

augmented by the biquadratic equation Dx{Dx(J.j+g.g)}.(Jf +gg) - 0-2Dy{Dy(f.f+g.g)}.(lf + gg) = 0 equivalent to the condition C{)xy = C{)yx'

The Hirota method is now used to construct a special class of solutions of the Ish­

I equation. In general, the functions f and 9 depend both on z = -~(x - iy) and2t

i = ~(x + iy) . However, let us first consider the case f = n», t) and 9 = g(i, t) . Such2t

restricted forms are compatible with equations (3.2.13) if

ift + ~f.zz = 0, (3.2.14)

The linear equations (3.2.14) admit the series solutions [257]

N

9 = s» = L L -¥t(2ii)m(2it)n,j=O m+2n=j m.n.

(3.2.15a)

(3.2.15b)

where aj and bj are arbitrary complex constants and N is an arbitrary integer. On

substitution ofthe expressions (3.2 .15) into (3.2.12), we obtain a family of exact rational

nonsingular solutions, namely, lump solutions of equation(3.2.1) [257]. For the simplest

one-lump solution (N = 1), one has II = ao, g1 = bo+ 2bdi, and

8'S 213(0: - (x - iy)) 213(x - xo - i(y - Yo))

1+t 2= =la+ x + iyl2 + 1131 2 Ix - xo + i(y - yo)12 + 11312'

83

= _Ia+ x + iyl2 - 1131 2 = Ix - xo + i(y - yoW - 1131 2

[o + x + iyl2 + 11312 Ix - xo + i (y - yo)12 + 1131 2 '

where a = -(xo + iyo) = -bo/b1and 13 = ao/b1' Equivalently,

. 2113 lpeiCl" - I"o)

8 1 + t82 = - p2+ 11312 '

where peil" := x - xo + i(y - Yo) and 13 = 113lei l" o.

(3.2 .16)

(3.2.17)

Other Integrable Equations and Methods in 2+1 Dimensions 121

Hence, the one-lump solution of equation (3.2.1) corresponds to a configuration of

vortex-type for the spin field 5(x, y, t) with center at the point (xo,yo). This solution is

static. The two-lump solution which corresponds to N = 2 and h = ao+ 2al iz, g2 =bo+ 2b1iz + bz (~(2iz)2 +2it) is, however, nonstatic and describes the collision of two

vortices of the type (3.2.17). The analysis of this solution shows that the collision of

the vortices is completely elastic and that the phase shift is absent as in corresponding

cases for KP-I, NVN-I+, etc .

The N-lump (N-vortex) solutions are topologically stable configurations of the spin

field 5. To convince ourselves of this fact, we calculate the topological charge Q (3.2.3)

for the N-vortex solutions. Taking into account (3.2.1) with a = i, one has

Q=-81

!!dXdy(<pxx + <pyy) =-81

lim !(<Pxdy-<Pydx).rr rr x 2+ y 2_ ooS

(3.2.18)

It follows from (3.2.12b) and (3.2.15) that, for the N-vortex solutions, <P = 2In(J!+99).

Hence, for these solutions <P ---+ 2ln IbN(2iz)2NI + O(!zl-l) as [z] ---+ 00 whence, on

substitution in (3.2.18), we obtain [257]

Q=N. (3.2.19)

Thus, the N -vortex solution of the Ish-I equation (3.2.1) is characterized by the

value N of the topological charge. It is for this reason that topologically nontrivial

fluctuations of the spin field can deform solutions with different numbers of vortices

into each other. Note that, by virtue of (3.2.18), the auxiliary scalar field <p{x, y, t)

plays the role of the potential created by the distribution of the topological charge1 - - -density -S.{Sx x Sy).

4rrOne can construct another family of exact solutions for the Ish-I equation by

choosing the functions! and 9 in (3.2.132) in the form! = !(z , t) and 9 = g{z, t}.The corresponding solutions differ from those constructed above by the change z ---+ -z,that is, by the substitution y ---+ -y in the expressions (3.2.15). The topological charge

for such solutions is Q = -N. Hence, these solutions are anti-vortex solutions.

In summary, it has been seen that the Ish-I equation (3.2.1) describes, in particular,

the time dynamics of spin vortices and anti-vortices of the form {3.2.17} on the (x,y)

plane.

3.3. The Manakov-Zakharov-Mikhailov equation

An elegant illustration of a nonlinear integrable equation which arises out of the

second method of the multidimensionalization as described in Chapter 1 is the Manakov­

Zakharov-Mikhailovequation [30, 261]

(3.3.1)

122 Chapter 3

where ~ = ~(x + ay) , .,., = ~(x - ay) , ge := 8eg, gT/ := 8T/f, 0'2 = ±1, and dctg = 1.

Equation (3.3.1) is equivalent to the compatibility condition for the two linear problems

[30,261]

()..8T/ - )..-18e- )..-IB + A)t/J = 0,

(8t + )..8T/ + A)t/J + 0,

(3.3.2a)

(3.3.2b)

where A = «'«. B = g-lge, and)" is a complex parameter. In the case 0'= i , equation

(3.3.1) was first considered by Manakov and Zakharov in [261]. For a = 1, equation

(3.3.1), with a redefinition of the independent variables, appeared in Mikhailov's paper

[30]. We shall refer to equation (3.3.1) as the MZM-I equation in the case a = i and the

MZM-II equation in the case a = 1. Remarkably, the MZM-I and MZM-II equations

were derived by totally different methods. Thus, MZM-I equation was obtained in [261]

as a stationary version of the four-dimensional self-dual equat ion (1.3.33) while the

MSM-II equation arose out of a two-dimensional nonabelian Toda lattice equat ion in an

infinite matrix order limit [30]. Equation (3.3.1) is invariant under the three-dimensional

Lorentz group [30, 261] .

The linear system (3.3.2) is not the only auxiliary linear system which leads to

equation (3.3.1). In fact, the system (3.3.2) is equivalent, for instance, to t he system

(8e + )"8t + B) t/J = 0,

()..8T/ + 8t + A)t/J = o.

(3.3.3a)

(3.3.3b)

The auxiliary linear system for t he MZM-I equation in the form (3.3.3) is the stationary

version (83 --> 0) of the linear system Llt/J = 0, L2t/J = 0 with operators L1 and L2 of

the form (1.3.34) corresponding to the self-dual Yang-Mills equation (1.3.33).

Different forms of the auxiliary linear problem are convenient for different purposes .

For the solution of the Cauchy problem for the MZM equation, it is more convenient to

use the system (3.3.2). On the other hand, the system (3.3.3) is the most appropriate

starting point for the investigation of the algebraic-geometric structure of equation

(3.3.1).

An important feature of equation (3.3.1) which is typical of the second method of

multidimensionalization is that the corresponding auxiliary linear problems (3.3.2) or

(3.3.3) contain only first order derivatives in the independent variables. This property is

key to the integration procedure of the MZMequation. Another important characteristic

of the MZM equation is that the auxiliary systems (3.3.2) and (3.3.3) contain the spectral

parameter Xexplicitly, in contrast to the auxiliary linear problems for (2+1)-dimensional

integrable equations considered earlier (see, for example, (2.1.2), (2.4.2), (2.6.21)).

Other Integrable Equations and Methods in 2+1 Dimensions 123

Thus, the system (3.2.3) is the analog of equations (2.1.4), (2.4.7), and notably, of

equation (2.6.3).

Let us consider first the MZM-I equation

The solution of the initial value problem is treated as in [261]. The corresponding

spectral problem is

(3.3.4)

(3.3.5)

where Z = 4(x + iy ). It will be assumed that 9 -+ 1 (that is, A, B -+ 0) as x 2 +y2 -+ 00. The simplest solution of equation (3.3.4) corresponding to A = B = 0 is

'l/Jo = exp[i(AZ + A-I z)]. It is bounded for all Z at IAI = 1. It is also noted that equation

(3.3.4) is invariant under the transformations 'l/J -+ 'l/J' = 'l/J f (AZ+ A-I z), where f is an

arbitrary function .

Following the approach considered in Chapter 2, we seek solutions X of equat ion

(3.3.4) bounded for all Aand canonically normalized (X -+ 1 at A-+ 00). Such solutions

can be represented as solutions of the integral equation

x{Z, s,A) = 1+JJdz' /\ dz'G(z - z', z - Z',A)U{Z',Z',A)X{Z', Z', A),

where u{z, z,A) := A-I B(z, z, t) - A{z, z, t) . The Green function G(z, z, A) of the

operator Lo = A8i - A-I 8z is readily calculated, and at IAI=f 1, one has

G{ - x) = 2- sgn (IAI- 1)z, Z, A 4 \ \ 1 •

1r AZ + A- Z(3.3.6)

The formulae (3.3.5) and (3.3.6) can also be derived by another method. Thus , let us

introduce the variable v := AZ + A-I Z. If we set A= IAlci'f', then

(3.3.7)

whereVI = z cose + y sin ip,

(3.3.8)V2 = -xsincp + ycos cp.

It may be verified that

In the terms of the variables V and v, equation (3.3.4) becomes

8X{V,V,A) 1 (\-1 A)8v = IAI2 _ IAI-2 A B - X·

(3.3.9)

(3.3.10)

124 Chapter 3

Accordingly, the linear problem (3.3.4) is representable in the form of a spatial 8·problem.

Application of the formula (1.4.10) in the case (3.3.10) with the assumption that

X -+ 1 as Ivi -+ 00 produces the integral equation

1 II dv' /\ dfi' u(v' fi' A)X(v,fi,A) = 1+ 27l'i IAI2 -IAI-2 (v,'- ~) X(v',fi',A)

G

(3.3.11)

equivalent to equation (3.3.5).

Let us investigate the analytic properties of the solution of equation (3.3.5) . TheGreen function G(z, Z A), as follows from (3.3.6), is ambiguous at IAI = 1. Hence, the

solution X has a jump across the unit circle IAI = 1. For all other A, the Green function

G is analytic . As a result , the solution X of equation (3.3.5) is analytic outside the unitcircle IAI = 1 except at points at which the homogeneousequation (3.3.5) has nontrivial

solutions. We will assume that the number of these points is finite and that all of them

are simple. Thus, the solutions of equation (3.3.5) are of the form

( s.x) ~Xi(Z,Z) ~(_\)XZ,Z,I\ =l+LJ A-A ' +xz,Z,I\,

i=l •

(3.3.12)

where Xis analytic for IAI ~ 1. The function Xis of the form X= {~~: l~: ~ ~ where

the functions X+ and x- are solutions of the integral equations

(3.3.13)

(3.3.14)

The boundary values of the solutions X± as IAI -+ 1 obey equation (3.3.13) with Green

functions G±(z, Z,A) which are the boundary values of G(z, z,A) as IAI -+ 1 ± 0,respectively. Hence, we arrive at a singular Riemann-Hilbert problem with the unitcircle IAI = 1 as the conjugation contour. The character of this Riemann-Hilbert problem

(local of nonlocal) can be determined by the method adopted in Chapter 2. Thus, we

derive the equation for the jump using the equality (G+ - G-)(e) = ~8(e) then express

the jump via another solution of (3.3.4) and so on. However, here it is simpler to use

equation (3.3.4) directly, or, rather, equation (3.3.10). As IAI -+ 1, by virtue of (3.3.7)

and (3.3.8), one has v = VI and (IAI2- IAI-2) 88_ -+ 2i88

. Therefore, for A= e- i rp

V V2equation (3.3.10) yields

8X(VI, V2, <p) 18 = -2·u(VI,V2,<P)X(VI,V2,<P) ,

V2 t

and it follows from (3.3.14) that

88 (Xll(Vt,V2,<P)X2(Vl,V2,<P)) = 0,V2

Other Integrable Equations and Methods in 2+1 Dimensions 125

where Xl and X2 are two arbitrary solutions of equation (3.3.14). In particular, choosing

the boundary values X+(VI, V2, cp), X-(VI, V2, cp) of the solutions X+ and X- as Xl and

X2, we get

that is,

(3.3.15)

where R(VI, cp) is some matrix function. Thus, the boundary values of the functions

X+ and X- are connected by a local relation and the conjugation problem under

consideration is a local Riemann-Hilbert problem.

The solution of the problem (3.3.15) is given by the standard equations (1.2.14) or

(1.2.15) on taking into account (3.3.12). The inverse problem data R(VI,cp) is simply

expressed via the asymptotics of the solutions X+, X- . Indeed, passing to the limits

V2 -+ ±oo in (3.3.15), we obtain

(3.3.16)

where

The experimentally observable scattering matrix S(Vl, cp) can also be expressed via the

asymptotics Xr±) (VI, cp) according to

(3.3.17)

Formulae (3.3.16) and (3.3.17) allow us to establish the relation between R(VI,cp) and

S(VI, cp) . This relation involves linear integral equations.

A more explicit expression for the scattering matrix can be found if one considers

the solution t/J(VI,V2,Cp) of equation (3.3.14) such that t/J ~ 1. Then, [261]V2-+-CJQ

= ;.1+00

dV~U(VI,V~,CP)t/J(v],v~,cp)= Pexp [;.1+00

dV~U(VI'V~,CP)] ,t -00 t -00

(3.3.18)

where P denotes the v2-ordered exponent. Accordingly, the mapping U(VI ,V2 , cp) -+

S(v], cp) can be regarded as a nonabelian Radon transform [2611.

Note also that, by virtue of the condition detg = 1, one has trA = trB = 1 so

that det S = 1. Further, it may be shown that if 9 is a self-adjoint matrix, then the

126 Chapter 3

matrix g-I[(X-(X-1))-I]+ is a solution of equation (3.3.4) along with the matrix X(A).

It follows that

(3.3.19)

and

The relation (3.3.19) in the limit A -+ 00 gives the simple reconstruction formula

g(z,z,t) = (X+(z,z, tj A= 0))-1. (3.3.20)

To solve the initial value problem for the MZM-I equation we must now determine

the temporal evolution of the inverse problem data. On substitution of the expression

(3.3.16) into (3.3.26) with ..\ = e-i tp, we find

Hence,

Analogously,

R(vl, ep, t) = R(VI - t, ip, 0).

S(Vl, ep, t) = S(VI - t, ep, 0).

(3.3.21)

Formula (3.3.21) and the inverse problem equations allow us to solve the initial

value problem for the MZM-1 equation by the standard 1ST scheme.

Now let us consider the solutions of the MZM-I equation which correspond to the

poles of the function X(..\). These poles generate the lump solutions of the MZM-I

equation. For the simplest solution of this type the function X(..\) is of the form [261]

_ Ao - AnI _X(z, z,A) = 1+ A_ Ao Q(z,z,t),

where the matrix elements of Q are given by

(3.3.22)

(i,k = 1, . .. ,N)

and the functions fi obey the equations

(8z + Ao8t )fi = 0,

(8t + Ao8i )f i = O.

Note that the matrix Q is a one-dimensional Hermitian projector (Q2 =Q).

(3.3.23)

Other Integrable Equations and Methods in 2+1 Dimensions

It follows from (3.3.12) that

127

(i=I, . . . ,N) (3.3.24)

where Vo = AOz + AOI Z and fi are arbitrary functions .

The solution of the MZM-I equation which corresponds to (3.3.22) is, by virtue of

(3.3.20), of the form

g(Z, z, t) = 1 + (IAol2- I)Q . (3.3.25)

On choice ofthe functions fi as rational functions fi(VO - t), we obtain a wide class of

rational solutions of the MZM-I equations.

In the simplest case, N = 2 with II = 1, h = f(vo - t), we have

where

_ IAol2 - 1 ( 1, f( vo - t) )g(z, z, t) = 1 + 1 + If(vo - t)12 l(vo _ t), Ifl 2 '

(3.3.26)

(3.3.27)

Here n and m are arbitrary integers and c, al, . . . , lln, bl , . .. , bm are arbitrary complex

parameters. The parameter Ao defines the velocity v = IAol +2!AOI- I with which the

solution (3.3.26) moves. This solution has no singularities on the (x, y) plane and

g(x, y, t) -+ const in all directions as x2 + y2 -+ 00.

The function X of the form (3.3.12) with n poles gives the solution of the MZM-I

equation which describes the collision of n lumps. As in the case of the lump solutions

of the KP-I, DS-I, NVN-I, and Ish-I equations, the lumps of the MZM-I equation do

not interact completely [261] .

Let us now move on to the MZM-II equation, that is, to equation (3.3.1) with

~ = Hx+y) and n = ~(x-y). The corresponding spectral problem is (3.3.2a) with the

real coordinates ~ and n (0" = 1). An analog of the complex variable v is given by the

variable v = A~ + A-I"." where A is a complex parameter. Putting A= IAle-i,!" one has

where

In terms of this variable,

VI = IAI~ + IAI - I"."

V2 = IAI- I"., - IAI~·

(3.3.28)

(3.3.29)

(3.3.30)

128

and equation (3.3.2a) becomes

(~ _ ~) at/!. = u.I..A A av '1/

Chapter 3

(3.3.31)

Hence, as in the case a = i, the spectral problem for the MZM-II equation is equivalent

to a spatial a-problem.

Let us consider the solutions X of equation (3.3.31) bounded for all vand normalized

as X -4 1 at Ivl -400. By virtue of (1.4.10), for such solutions, one has

(

A ';; A) _ 1 II dv' Ad~' u(V',~',A)X(V',~',A)X v, v, - 1 + -2. ( -)' (A' A)ll't A A V -v

c X-X

or, in terms of the variables e, TI,

(3.3.32)

It follows from the integral equation (3.3.32) that the function X(e, TI, A) is analytic

at ImA f 0, ReA f O. For real and pure imaginary A, the integrand in (3.3.32) is

ambiguous and, consequently, the function X has a jump across the real axis ImA = 0

and imaginary axis ReA = O.

Thus, as for the MZM-lequation, we have a Riemann-Hilbert problem, but now the

conjugation contour is the union of the real axis ImA = 0 and imaginary axis ReA = 0

instead of the unit circle IAI = 1. This Riemann-Hilbert problem is local. This can be

readily established directly from (3.3.10) . At 1m>. -40, one has (X - ~) :~ -42 a:2and (X - ~) :v -4 2a:

1at Re>. -4 O. So equation (3.3.10) becomes

Imx = 0,

(3.3.33)

ReA =0.

(3.3.34)ImA = 0,

These equations imply, in a manner similar to the case a = i, that the boundary values

xt,xl and Xt,X2 of the function X as Imx -4 ±O and RCA -4 ±O are connected by

the relations

Xt('01, '02, >') = xt('Ol, '02 ,>')R2('02, >'), Rc>. = O.

Finally, the time-dependence of the inverse problem data Rt('Ot ,>., t) and R2('02, >., t) is

of the form

(3.3.35)

Other Integrable Equations and Methods in 2+1 Dimensions 129

This allows us to solve the Cauchy problem for the MZM-II equation in the standard

way. These lump-type solutions of the MZM-II equation have been constructed in [418] .

Thus, we have seen that the linear auxiliary system (3.3.2) allows us to solve the initial

value problem for the MZM-II equation. The linear auxiliary system (3.3.3) turns out

to be convenient for the study of other properties of the MZM equation.

Let us first consider the MZM-I equation and introduce the complex variables

t - >.zv =----x-' >.t - z

w=->.-. (3.3.36)

In the terms of these variables, equations (3.3.3) adopt the form

8'1/1(v,u,w,w) (_) (_ _)8u + B z,z, t '1/1 v,v,w,w = 0,

8'1/1(v,v,w,w) A( - ) (- -)8w + z, z,t '1/1 v, v,w,w = 0,

(3.3.37)

that is, they comprise a pair of spatial a-equations. The compatibility condition for the

system (3.3.37) is8A 8B- - - - [A B] =0.8v 8w '

(3.3.38)

(3.3.39)

This is the zero a-curvature condition for two-dimensional complex space or, equiva­

lently, the a-closure condition for the one-form n = Adw + Bdv.

Equation (3.3.38) is equivalent to equation (3.3.1). Moreover, the formulae

B(z, z,t) = - ~~ .'1/1 - 1,

A(z, z, t) = - ~~ .'1/1 - 1

which follow from (3.3.37) or (3.3.38), give the solution of equation (3.3.1) with A =

g-lgt , B = g-lg",, for any function 'I/1(v ,v, w, w) such that the right-hand side (3.3.39)depends only on the variables z, z, and t.

We see that the MZM-I equation has a simple geometric interpretation in that the

manifold of solutions of the MZM-I equation is equivalent to a certain subspace of the

two-dimensional complex space (v,w) with vanishing a-curvature. We will not specify

this subspace here, but this type of geometric interpretation will be discussed for the

self-dual Yang-Mills equations in Chapter 5.

For the MZM-II equation, the analogs of the variables (3.3.36) are of the form

_ t - >.~v=---,>.->.

_ T/ - >.tw= >._>- . (3.3.40)

The formulae (3.3.37)-(3.3.39) are valid for the MZM-II equation with the substitutionsv ..... v, w ..... w.

130 Chapter 3

In conclusion, it is noted that the MZM-II equation (3.3.1) can be rewritten in the

form [418]

(J-L = 0,1,2) (3.3.41)

where x P = (xO,x1,x2), op == ojoxP , -yP" = diag (1,-1,-1), fo.P" is the full anti­

symmetric tensor (f012 = 1), and k = (kl, k2,k3) is a constant unit vector . The choice

k = (1,0,0) gives the MZM-I equation, while for k = (0,1,0), equation (3.3.41) is

equivalent to the MZM-II equation. In the latter case, equation (3.3.41) possesses the

conserved energy-momentum tensor of the form [418]

The form (3.3.41) manifestly demonstrates the distinction between the MZM equation

and the three-dimensional chiral fields model equation (k = 0).

3.4. Nonlocal, cylindrical, and other generalizations of the

Kadomtsev-Petviashvili equation

In this section, we discuss certain nonlinear integrable equations which are closely

connected to the KP equation. We start with a nonlocal generalization of the KP

equation in the form [419, 420]

Ut + 2(at5)-1 HD 2u+ 2uDu - 4(at5)-2Du = 0, (3.4.1)

where u(x, y, t) is a real scalar function, 15 is a real parameter, a2 = ±1, Du := U x ­

uy, H:= (T + I)(T _1)-1, and T is the shift operator (TJ)(x) := f(x +ac). Equation

(3.4.1) is equivalent to the compatibility condition for the linear system

1(Oy - Ox + T + "2at5u)'l/J = 0,

(Ot + 2(ac)-lT2 - ivT + w)'l/J = 0,

(3.4.2)

where v = i(Tu +u + 2(O'C)-2), W = Hux - 2(O'C)-lu - O'u2.

In the limit C -. 0, equation (3.4.1) reduces to the KP equation on appropriate

redefinition of the independent variables . The transition from the KP equation to

equation (3.4.1) (the NKP equation) is analogous to the transition from the KdV

equation to the intermediate long wave equation.

We now introduce another important technique for the construction of exact so­

lutions to integrable nonlinear equations. This is the Darboux transformation method

which represents a generalization of observations originally contained in [421] and which

was subsequently developed, in particular, by Matveev in [422, 423]. This method

Other Integrable Equations and Methods in 2+1 Dimensions 131

is based on the use of deep symmetry properties of the auxiliary linear problems

which involve their invariance under certain transformations of both the potentials and

eigenfunctions.

The application of the procedure to equation (3.4.1) is based on the form-invariance

of the system (3.4.2) under the Darboux transformations [419, 420]

1/J --+ 1/J' = T1/J - T~l .1/J,

, 2 (T1/Jl)U --+ U = U+ O'68x In ~ ,

(3.4.3a)

(3.4.3b)

where 1/Jl is a fixed solution of the system (3.4.2) with the potential u(x ,y , t) . One can

check directly the invariance of the system (3.4.2) under the transformations (3.4.3). An

obvious consequence of this invariance is that u'(x, y, t) is a solution of equation (3.4.1)

together with u(x, y, t). Thus, given one solution u(x, y, t) of the NKP equation (3.4.1),

on calculation of the corresponding 1/J and 1/Jl and insertion in (3.4.3b), a new solution

u'(x, y, t) of the NKP equation can be constructed. On repetition of the Darboux

transformation N times, we obtain a solution of the NKP equation in the form [419,

420]1

UN(X, y, t) = u(x, y, t) + 0'6 (T - 1)8x ln det AN, (3.4.4)

where (AN)ik = Ti-l 1/Jk (i,k = 1,... ,N ) and 1/J l ,1/J2 , .. . , 1/JN are solutions of the

system (3.4.2). In this way, the Darboux transformation method allows the construction

of infinite families of solutions of the NKP equation starting from one seed solution.

The simplest starting solution corresponds to the choiceof the trivial solution u == O.

The elementary solution of the system (3.4.2) with u == 0 is

1/Jo(x, y, t) = ekx+a(k)y+b(k)t,

where k is an arbitrary constant and

a(k) = (O'~)2 (eu6k

- O'6k - 1),

b(k) = _2_(2eU6k _ e26uk)(0'6)3 .

(3.4.5)

(3.4.6)

A single application of the Darboux transformation is sufficient to give a rich family of

solutions of the NKP equation [419, 420]. Indeed , let us choose the solutionu, in the

formN / +001/JJ (x, y, t) = {;Ci1/JO(X,y, t ,ki ) + - 00 dkp(k)1/Jo(x, y, t ,k). (3.4.7)

132 Chapter 3

For positive Ci, p(k), and real ki, the formula (3.4.3b) provides real nonsingular solutions.

In the case p == 0, these solutions decrease exponentially as x ---> ±oo for fixed y and t .

On the other hand, if one chooses the function '1/11 as

'I/11(X,y,t) = L fOll0l20l30l4(X)8~1a:28r38k4'1/1o(x,y,t,k),Q1, .. · , Q 4

then the formula (3.4.3b) gives a family of rational solutions of the NKP equation

which decrease as x-2 at x ---> ±oo. Solutions dependent on arbitrary functions can be

constructed via the formula (3.4.4) if one takes the functions 'l/Ji in the form (3.4.7) with

different pi(k) .

Other classes of exact solutions of the NKP equation can be generated by this

method. These include, in particular, periodic and quasi periodic solutions which

generalize the solutions of the KP equation in the form (2.3.43) [419, 420) . In the

limit 0 ---> 0, all the above mentioned solutions reduce to solutions of the KP equation.

Application of the Darboux transformation method to the construction of wide classes

of exact solutions of a wide variety of integrable nonlinear equations is described in

[421-423] .

We next consider the cylindrical KP equation (the Johnson equation) descriptive of

surface water waves for small deviation from axial symmetry. It adopts the form [424)

where a 2 = ±l.

(V ) 2Vyy

Vt + Vx x x + 6vv x + 2t x + 3a t2 = 0, (3.4.8)

It has been shown in [425, 426] that the 1ST method is, in fact, applicable to

equation (3.4.8) with operators £1 and £2 given by

2 X£1 = a8y - t8x - tv + 12'

£2 = 8t + 48; + 6v8x + 3(vx + T8;1Vy) .(3.4.9)

Equation (3.4.8) is equivalent to the condition [£1, £2] = 0. Particular solutions of

equation (3.4.8) have been constructed by the Darboux transformations method in [425].

The initial value problem for the cylindrical KP equation (3.4.8) for decreasing

v(x, y, t) has been studied in [427] via methods already described in Chapter 2. It

turns out that for equation (3.4.8) with a = i a nonlocal Riemann-Hilbert problem

results while for equation (3.4.8) with a = 1 a a-problem may be constructed. Rational

nonsingular solutions (lumps) have also been found for the cylindrical KP-I equation

[427].

Other Integrable Equations and Methods in 2+1 Dimensions 133

The most important property of equation (3.4.8) is, however, the correspondence

between the cylindrical KP and KP equations. Thus, if u(x , y, t) is a solution of the KP

equation, theny2t

vex, y, t) := u (x - 120'2' yt , t) (3.4.10)

is a solution ofthe cylindrical KP equation (3.4.8) [428]. This relation is invertable, and

hence, a solution vex, y, t) of equation (3.4.8) generates via

y2 yu(x,y,t) = v(x+ 120'2t' t,t) (3.4.11)

a solution of the KP equation. This connection between the KP and cylindrical KP

equations is the two-dimensional generalization of an analogous relation between the

KdV and cylindrical KdV equations [429]. This connection is reflected in a one-to­

one correspondence between all stages of the solution of the initial value problems forthe KP equation (Sections 2.1 and 2.2) and the cylindrical KP equation [427]. It alsoallows us to solve the Cauchy problem for equation (4.3.8) by simple reference to the

corresponding solution of the KP equation [428]. Indeed, given vex, y, to) := vo(x, y),

from (3.4.11) we have u(x, y, to) = uo(x, y) = v (x +4, JL, to). Then, solution120' to to

of the Cauchy problem for the KP equation (Sections 2.1, 2.2) gives u(x, y, t) for t > to.

Finally, using formula (3.4.10), we may construct v(x,y,t) for arbitrary t > to.

An elegant illustration of the use of the above correspondence is provided by the

construction of the N-lump solutions VN of the cylindrical KP-I equation directly from

the N-lump solutions UN of the KP-I equation via

VN(X ,y,t) =UN (x+ ~;, yt,t) ,

where the functions UN are of the form (2.3.3). Moreover, by virtue of (3.4.10), allsolutions of the KP equation as constructed in Section 2.3 may be mapped into solutionsof the cylindrical KP equation.

It is noted that the relation (2.4.10) also allows the Hamiltonian structure of the

cylindrical KP equation to be determined via the known Hamiltonian structure of theKP equation [428] .

Another equation connected with the KP equation is the nonlocal analog of the

KP equation considered in [430] , namely,

(3.4.12)

where vex, y, t) is a scalar function, 0'2 = ±1, and HI is the Hilbert transform of the

function I given by (HJ)(x) = ! 1+;' ~(X') . It is readily verified that, if the function7r -00 X - X

vex , y, t) obeys equation (3.4.12), the function

u(x, y, t) := (1+ iH)v (3.4.13)

134 Chapter 3

obeys the KP equation [430]. The correspondence (3.4.13) can also be generalized toother types of integral operators [430].

Finally, we note that there is also a supersymmetric generalization of the KP

equation [431] .

3.5. The Mel'nikov system

The auxiliary linear systems considered up to now have allowed detailed investiga­tion of the associated integrable equations . Further complication of the spectral problemnecessarily complicates subsequent analysis. However, the study of more complexspectral problems (involving matrices, higher order derivatives etc.) is desirable sincereductions of such systems can generate new interesting (2+ I)-dimensional integrablesystems.

A wide class of complicated linear systems has been studied by Mel'nikov in a seriesof papers [268, 432-438]. Therein, the triad representation (1.3.21) notion was used toconsider a class of nonlinear systems which possess the representation [LI , L2] = BLIwith operators L1 and L2 of the form [433]

L-08y , VI ,V2 ,· ··, VN

(3.5.1)

(3.5.2)

(3.5.3)

WI, 8x,0, . . . , °LI= W2, 8x,0, .. . , °

W2 , 0,8x , ' " , °WN , 0,0, ... , 8x

Qo ° °QI ° °L2 = 8t +

QN ° °n

L = 8;+1 + L Uk(X, y, t)8;,k=O

and

where

n

Qo = 8;:+2 +L qk(X, y, t)8; ,k=O

and Vk, Wk , Qk, (k = 1, . . . , N) are differential operators of order m + 1. The operatorB has the structure

B= (3.5.4)

Other Integrable Equations and Methods in 2+1 Dimensions 135

where Pk and Qk (k = 1, . . . , N) are differential operators. The operator equation

[£1l L 2] = BL I is equivalent to a system of the differential equations for the functions

Uk,Vk,Wk·

The operators L l and L2 in the form (3.5.1), (3.5.2) represent a special matrix gen­

eralization of operators of the type (1.3.1). Consequently, the integrable systems which

correspond to the operators (3.5.1) and (3.5.2) are generalizations of the corresponding

integrable scalar equations.

We consider below in (i)-(iii) the three simplest examples.

(i) n = m = 0, a = 1, L = Qo = a; + u, U = U, WI = WI := iv, QI

wax + w x, PI = -vax - 2vx. The corresponding integrable system is [437]

Ut - u y + 2(lvI2)x = 0,(3.5.5)

iVt - uv - Vxx = o.

This system describes the interaction of a long wave with a packet of short waves

which propagate in nonaligned directions in the plane. The system (3.5.5) is obviously

equivalent to the system (1.3.8), that is, to the degenerate DS equation (at - ay -+ a,.,).

(ii) n = m = 1, a = 1, L = Qo = a~ + uax + !ux, VI = WI == v, PI =

-va; - 3vxax - 3vxx - uv, QI = -va; + vxax - Vxx - UV. The nonlinear integrable

system in this case is of the form [434]

Ut + u y - 3{v2)x = 0,

(3.5.6)

This system describes the interaction of two types of long waves in the (z, y) plane. On

introduction of the variable z defined by az := at + ay from (3.5.6) ,

J:)-I( 2) 3 J:)-I{ 2)Vt + vxxx + 3vxuz v x + '2vuz v xx = O.

This is a (2 + I)-dimensional generalization of the mKdV equation (1.1.9).

(iii) In this case, we choose the operators L 1, L2 , and B in the form

(3.5.7)

-ivax - 2Vx)

o '

136

and arrive at the system [437]

4Ut - '3 p% = 0,

1 1 2 12 )Pt - :4UY - :4 (u%% + 3u + Iv % % = 0,

iVt - uv - v%% = O.

On elimination of P from (3.5.8a) and (3.5.8b) , we obtain the system

3Utt - (uy + U u %+ 6uu% + 8Ivl~)% = 0,

iVt - v%%- UV = o.

Following the substitution y +-+ t, the system (3.5.9) becomes

ivy - Vu - uv = 0

Chapter 3

(3.5.8a)

(3.5.8b)

(3.5.8c)

(3.5.9)

(3.5.10)

and describes the interaction of long (u) and short (v) waves in the (x, y) plane. For

v = 0, we obtain the KP equation.

The operator representation for the system (3.5.10) is

where

A ( 8; + u, 0)L 1 = -i8y + _ '

-v% + v8%, 0

and

(

0,B=

v8% - v%,

-v8%0- 2v% ) .

Note that the KP equation (that is, t he system (3.5.10) with v == 0) possesses a commu­

tativity representation, while for the full system (3.5.10), a triad operator representat ion

is required.

The multicomponent versions (N > 1) of all the above equations are given in [433] .

It has already been seen that integrable equations in 2+ 1 dimensions possess rich

families of soliton-type solutions which describe various nontrivial processes (resonant

Other Integrable Equations and Methods in 2+1 Dimensions 137

interaction, decay, etc). The equations introduced in this section have analogous solu­

tions . These have been constructed in [268, 435-4381 via a procedure markedly different

from the standard 1ST method.

First, let us consider the system (3.5.5). The simplest soliton-type solution of

(3.5.5) is of the form [437]

2JL2U(X, y, t) = 2 ,

cosh [JL(x + 2vt + 2XY)]

( )exp[iv(x + 2vt) + iTY]

v x y t = c-=:-':----7----,:----::..:,., , cosh[JL(x + 2vt) + 2XY] ,

(3.5.11)

where the real parameters JL , v, X and the complex parameter c obey the constraint

2(v - X)JL2 + Icl 2 = 0 so that X - v 2:: 0 and T is an arbitrary parameter.

The general N-soliton solution of equation (3.5.5) is given by the formula [437,4381

u = 28; Indet A,detB

v = detA' (3.5.12)

where the 2N x 2N matrix A and 3N x 3N matrix B are given by

(3.5.13)

Here I is the N x N identity matrix, while A and Rare N x N matri ces with elements

where

and

ZRmn =;;2 2 '

I'n - Pm(m,n= 1,2,oo.,N) (3.5.14)

2 -2 2'Pm = Tm + Wm + ZJLmXm ,

8m = vmX+ TmY + (v~ - JL~)t.

Here JLm , Vm, Xm, Tm. are real quantities, f = (1, . . . , 1) and A= (Al , . . . , AN)T.

Under the additional restrictions sgn JLl = sgn JLN , Xm - Vm > 0 (m = 1, . . . , N) ,

it follows that det A > 0 for all x ,Y, t , and hence, the solution (3.5.12) has no singularities

for all x , y, and t.

The expression (3.5.12) describes the interaction of N solitons of the type (3.5.11).

The interaction properties depend critically on the interrelation between the constants

138 Chapter 3

Wm, P~, and Xm' When these quantities are all distinct, the solution (3.5.12) describes

the collision of N solitons of the type (3.5.11) each with their own parameters J.Lm, Vm, Xm

which obey the constraints 2{vm - Xm)J.Lm + leml2 = 0, (m = 1, . .. ,N). The solitons

have unaltered profiles following the collision and suffer only a phase shift . When some of

the parameters Wm,~,Xm coincide , complicated processes of gluing, decay, or capture

of solitons may arise [437, 438]. For instance, for N = 2, when (Wl -W2)(p¥ - p~) = 0 and

Xl = X2, the solution (3.5.12) with {Xl- X2)J.L2 > 0 has asymptotics ofthe form (3.5.11)

as y -+ -00, while as y -+ +00, the functions u and v vanish [436] . These asymptotics

are interchanged when {X2 - Xl)J.L2 < O. Further, if (Wl -W2)(p¥ - p~) = 0 and Xl = X2,

but Vl ~ V2, then the solution (3.5.12) describes various transition processes of two

solitons one into another [436] . For N > 2, more complicated processes can obtain [437,

438].

Analogous solutions also exist for the system (3.5.1O) . The simplest soliton-type

solutions in this case consist of the traveling wave [435]

(3.5.15)

v=O,

that is, the usual KP single soliton, and the stationary soliton

(3.5.16)

where the real parameters J.L2, V2 and the complex parameter c satisfy the constraint

Icl2 + {J.L~ - 3v~)J.L~ = 0, J.L~ < 3v~.

The solution of equation (3.5.10) which describes the interaction of solitons of the

two types (3.5.15), (3.5.16) has the form [435]

where

Bv=­

A'(3.5.17)

(3.5.18)

and

a> 0, (3 =

Other Integrable Equations and Methods in 2+-1 Dimensions 139

For 3v~ > J.L~, the solution (3.5.17) has no singularities.

The properties of the above solution depend on the relationship between WI and

W3· In the case WI f W3 with J.LIri > 0, the asymptotics of the solution (3.5.17) as

t -+ -00 deliver the soliton

(3.5.19)

v=O

and the soliton (2.5.16) with the substitutions c -+ c, - x -+ x + Xo-. As t -+ +00, the

solution (3.5.17) again delivers the soliton (3.5.14) but with the change L -+ 6+ and the

soliton (3.5.16) with the changes c -+ c+, X -+ X+ xo+. Thus, for WI f W3, the solitons

(3.5.15) and (3.5.16) collide elastically. Their profiles are unaltered asymptotically and

-1 IWI - W3/ 'Ythe phase shift is equal to xo+ - Xo- = -In _, Ic+ I = Ic-I and 6+ = ! In -(3,J.L2 WI +W3 I

1L = 21na [435].

The nature of the interaction changes crucially in the case WI = W3, that is, when

J.LI = J.L2 = J.L, VI = 112 = u, In this case, with J.L'TI > 0, the solution (3.5.17) produces

the traveling wave (3.5.19) asymptotically as t -+ -00 and the stationary wave of the

form (3.5.16) with the substitutions c -+ c., = (3tc, x -+ x + 2-ln (3 asymptotically as2J.L

t -+ +00. Hence, for WI = W3, the solution (3.5.17) describes the process of transition

of the soliton (3.5.15) into the soliton (3.5.16) and vice versa.

One can analyze, in a similar manner, the interaction of two solitons of the form

[268]

ei v (x+ 2vy )+i Xt . 2 2V = c e-'(/-l +v )y

cosh[J.L{x + 2vy) - 'Tt] ,

where the real parameters J.L, v, 'T and the complex parameter c obey the constraint

['T - 4{J.L2 - 3v2)]J.L2 - 41cl2 = O. As in the previous case, elastic collision, decay, and

gluing processes are possible [268]. In addition, boomeron-type processes of soliton

reflection can also arise [268].

2J1,2U - -----,.,----'----

- cosh2[J.L(x + 2vy - 'Tt)] ,(3.5.20)

It has been seen that equations (3.5.5) and (3.5.10) possess rich families of exact

solutions descriptive of various soliton interaction processes. This is typical of (2 + 1)­

dimensional nonlinear integrable equations. The lump solutions in 2+ 1 dimensions are

analogs of one-dimensional solitons. The lumps correspond to the discrete spectrum

(bound states) and their collision is relatively trivial. However, soliton-like solutions

in 2 + 1 dimensions are associated with the continuous spectrum and their properties

140 Chapter 3

are similar in many respects to the properties of the continuous spectrum solutions

(associated with radiation) in 1 + 1 dimensions.

3.6. The modified Kadomtsev-Petviashvili and Gardner equations.

.The Miura transformation and gauge invariance

The cylindrical KP equation, considered in Section 3.4, is one example of an

integrable equation linked to the KP equation by a nontrivial change of the independent

variables. Links between two integrable equations which involvechange of the dependent

variables are more common. The best known such example in 1 + 1 dimensions is the

Miura transformation [439] which connects the mKdV equation (1.1.9) to the KdV

equation (1.1.1).

An analog of the Miura transformation exists in 2+1 dimensions. The correspond­

ing modified KP (mKP) equation is [250]

(3.6.1)

The latter equation is equivalent to the commutativity condition for the two operators

L 1 = aay + a; + v(x, y, t)8x, (3.6.2)

(3.6.3)

It can be readily verified that, if the function v{x, y, t) satisfies the mKP equation, then

the function u{x, y , t) defined by

1 1 2 1 -1u{x Y t) = --v - -v - -aa v" 2

x42 x Y (3.6.4)

obeys the KP equation [250]. The relation (3.6.4) represents a two-dimensional gener­

alization of the Miura transformation.

The Miura transformation can be viewed in different ways. The natural derivation

arises out of a gauge invariant formulation of the KP equation [250, 440].

Let us start with the following auxiliary linear problems

L 21/J = [at + 4a~ + 6v~a; + (3Vlx + ~v~ - 3a;IV1y + 6vo)ax

+ 4vox + 2VOVI - 'P]tJ1 = 0,

(3.6.5a)

(3.6.5b)

Other Integrable Equations and Methods in 2+-1 Dimensions 141

where vo(x, y , t), VI (x, y , t), and <p(x, y, t) are scalar functions. The compatibility con­

dition for the system (3.6.5) is equivalent to the system of two nonlinear equations

VOt + 6vovo% + 2(vovdy - 4vo%y - 2VOV1%% - 2VOVI Vlx

1 2 s:'- VoxVlx - 2V1 VOx - 3vo% % Vly + <Py + VI <p% + vo<Pxx = 0,

vlt - 2vo%% - 6vOy + Vlxx% + 2(VOVl)%

3 2 a-I a-I- 2V1 Vl% - 3Vl% % Vly + 3 % Vlyy + 2<p% = O.

(3.6.6a)

(3.6.6b)

The presence of the arbitrary function ¢ in (3.6.6) corresponds to a nonuniqueness in the

form of the integrable systems which reflects the gauge freedom inherent in the auxiliary

linear system. In the present case, the system (3.6.5) admits the gauge transformations

'l/J(x, y, t) -+ 'l/J'(x, y, t) = g(x, y, t) 'l/J(x, y, t), (3.6.7)

where g(x, y, t) is a scalar function such that 9 -+ 1 as x2+ y2 -+ 00. The system (3.6.6)

is form-invariant, that is, it has the same form in terms of the new variable 'l/J', vb,v~,and <p', where

(3.6.8a)

(3.6.8b)

and

(3.6.9)

with tjl(VQ, VI) an appropriate combination of Vo and VI. One can easily verify that

the transformations (3.6.8) with different 9 form an infinite-dimensional Abelian group.Such infinite-dimensional gauge groups arise in gauge field theories (see, for example,

[441]).

The configuration space of the system (3.6.6) is parameterized by the two functions

vo(x, y, t) and Vl(X, y , t) . The gauge group with one functional parameter g(x, y, t) acts

on this configuration space. Consequently, only one combination of these two functions

Vo and VI represents a dynamical (nongauge) variable, while the other corresponds to a

degree of freedom in the gauge.

The most widespread method of elimination of gauge degrees of freedom consists in

fixing the gauge by imposing some additional constraints on VQand VI . In fact, in all the

cases we have discussed, the spectral problems have been in a fixed gauge. For the KP

equation, we used the gauge VI = 0 (see (1.2.3)). It follows from (3.6.8) that the gauge

freedom disappears under this restriction. The gauge condition VJ = 0 is not the only

one possible; one could also set Vo = O. The general linear gauge is aovo + al VI = 0,

where ao and al are arbitrary real constants. Employing the gauge transformation

142 Chapter 3

(3.6.8), one can convert any given linear combination vo and VI into zero. For instance,

with the choice 9 = exp a8;IvI), we obtain v~ = o. While, with the solution of the

equation (8y+ 8~ + vI 8x + VO)g-I = 0 as g, we arrive at vb = o. The two different

gauges are related to each other. Moreover, it follows from (3.6.9) that one can convert

any function cp into any other prescribed function cp' by a gauge transformation. This

means that equations (3.6.6) with different cp are gauge equivalent to each other.

The system (3.6.6) gives different equations corresponding to various gauges. For

instance, putting VI = 0, from equation (3.6.6b), we find cp = VOx +38;Ivoy. Substitu­

tion of this expression for sp into equation (3.6.6a) with VI = 0 gives the KP equation

for Vo. On the other hand, choosing Vo = 0 and correspondingly cp = 0, one gets from

(3.6.6b) the mKP equation (3.6.1) with a = 1. Since these two gauges are connected

by a gauge transformation, it follows that the mKP and KP equations are connected

by the same gauge transformation. Later, we will show that this gauge transformation

is nothing but the Miura transformation (3.6.4).

The second method of elimination of gauge freedom consists in the explicit extrac­

tion from Vo and VI of the dynamical and pure gauge degrees of freedom. To do this,

we first note that the function

(3.6.10)

is gauge invariant, that is, w(vb, vD = w(Vo, VI) under the gauge transformations (3.6.8)

[250]. This invariant represents a pure dynamical variable which is insensitive to gauge

transformations. The existence of this invariant indicates the possibility of extracting

the invariant and pure gauge degrees of freedom from the two functional degrees of

freedom implicit in Vo and VI . Indeed, the variables Vo and VI admit the following

parameterization [440]

(3.6.11)

where W is the gauge invariant variable and p is the pure gauge variable, that is,

w-w' =w, p- p' =gp (3.6.12)

under the transformations (3.6.8). For w = 0, the variables Vo and VI are of a pure gauge

type, that is, they can be converted to zero by the appropriate gauge transformation

(3.6.8). Thus w corresponds to a dynamical degree of freedom.

The aforegoing allows us to formulate an approach which is essentially different

from that of fixing the gauge. This second approach consists in the extraction from the

system (3.6.6) of the equation for the dynamical variable wand the equation for the

Other Integrable Equations and Methods in U1 Dimensions 143

pure gauge variable p. Thereby, one can verify that the system (3.6.6) is equivalent to

the equation

Wt + Wxxx + 6wwx + 3a;lWyy = 0 (3.6.13)

together with the equation for the gauge variable p.

Equation (3.6.13) represents the gauge invariant subsystem of (3.6.6). The dynam­

ics inherent in the system (3.6.6) are incorporated therein. Equation (3.6.13) is nothing

but the manifestly gauge invariant form of the KP equation. It is noted that one can also

derive equation (3.6.13) by excluding the function cp from the system (3.6.6) without

fixing the gauge.

We now return to the first approach wherein the gauge is fixed. It has been noted

that, in the gauge VI = 0, one has W= Vo and equation (3.6.13) is the usual KP equation.

In the gauge Vo = 0, equation (3.6.13) is equivalent to the mKP equation (3.6.1) for VI

(with a = 1). In view of the coincidence of W in these gauges (vc == U, VI == 0) and

(vo == 0, VI = V) one gets

(3.6.14)

This is precisely the Miura transformation (3.6.4).

Further, let us consider the general linear gauge aovo +a1 VI = 0 and introduce the

function v{x, y, t), such that Vo = /3ov, VI = /31 V, where ao/3o +a1/31 = O. In this gauge,

equation (3.6.13) is equivalent to the following

This equation is satisfied if

(3.6.15)

Equation (3.6.15) represents a mixed KP and mKP equation, namely, the (2 + 1)­

dimensional Gardner equation [440]. It is equivalent to the commutativity condition for

the operators

L1 = ay + a; + /31 vax + /3ov,

L2 = at + 4~ + 6/31va; + [3/31Vx + ~/3~v2 + 6/3ov - 3/31a;lvy] ax

+ 3/3ovx + ~/30/3lv2 - 3/3o(a;lVy) .

144 Chapter 3

The Painleve properties of equation {3.6.15} have been studied in [443].

The transformation which relates the Gardner equation {3.6.15} to the KP equation

is simply the gauge transformation from the gauge {vo = /3oV, VI = /3IV} to the gauge

{vo = U, VI = O}. From the equality w{v6, vD = w{vo, vd, we obtain

{3.6.16}

The transformation {3.6.16} is the two-dimensional generalization of the Gardner trans­

formation.

Note that similar gauge-free and manifestly gauge invariant formulations also exist

for other integrable nonlinear systems and, in particular, for the system linked to spectral

problems with {8y + E~=l vn{x, y, t}8r;}1/J = 0 [440].

It is remarked that the gauge transformations and equivalences considered in this

section are of an elementary kind. The gauge equivalence between the DS and Ishimori

equations as discussed in Section 3.2 is of a more complicated nature.

3.7. Further integrable equations in 2 + 1 dimensions

To conclude this chapter, we present further interesting examples of integrable

systems in 2 + 1 dimensions. We start with integrable equations associated with the

two-dimensional problem {p{x,y, t}8y + E~=o un{x, y, t}8r;}'lj; = O. The hierarchies ofcorresponding integrable equations have been constructed in [443] . We present here

only the three simplest equations of this type.

The first equation is [57, 433]

and is equivalent to the operator equation [L1, L2] = 0, where [443]

{3.7.2}

Equation {3.7.1} is the {2+ 1}-dimensional integrable generalization of the Sawada­

Kotera equation [444].

The second example is the two-dimensional generalization of the Kaup equation

[57, 443, 445]

Other Integrable Equations and Methods in 2+1 Dimensions 145

representable in the form [L1' L2] = 0 with [443]

1L1 = ay + a; + uax + 2ux,

(3.7.5)

(3.7.4)5 n3 45 2 (35 2 -1)L2 = at - 9ax -15uox- "2uxax - "2u:z::z: + 5u - 5(ax uy) ax

5- 5uux + 2'Uy - 5uxx.

Equations (3.7.1) and (3.7.4) can be rewritten in a purely local form by introduction of

the potential q defined by u = qx. The inverse problem for the general linear equation

(ay + :L~=oUna;) 'l/J = 0 as well as particular cases of equation (3.7.1) have beendiscussed in [446].

The third example is the equation [443]

3 -1 ( 2a-1(Uy ) )Ut + u Ux:z::z: + 3u u x u2 y = 0,

equivalent to the commutativity condition for the two operators [443]

L 1 = ay + u2(x, y, t)a;,(3.7.6)

(3.7.7)

L2 = at + 4u3a; - [6u2Ux - 6u2 (a;1 C~))] a;.

In the terms of the variable sp := u-2 , equation (3.7.5) becomes

CPt - 2 ((cp-!):z::z:x + 6cp2 (cp-1a;1 (cp!)Jy = O.

Equation (3.7.7) represents the (2+ l)-dimensional integrable generalization of the (1+

l)-dimensional Harry Dym equation CPt - 2 (cp-!) = 0 (see, for example, [100]) .xxx

Novel invariance properties of equation (3.7.7) and its relation to the KP equation have

been discussed in the paper [447]. The Painleve properties have been considered in[442].

(3.7.8)

(a - --!!:!!:-ax) 8 = - 2na QR,u m+n n+m

There is considerable interest in not only scalar integrable equations but also in

their multicomponent, matrix, and other generalizations. In [449], generalizations of

the KP, DS, and N-wave equations associated with symmetric and homogeneous spaceshave been constructed. One such generalization of the DS equation to symmetric space

adopts the form [449]

mna2aQt + Qyy + ( )2 Q:z::z: - 2QRQ + 8xQ - QTx = 0,

m+n

mna2aRt - Ryy- ( )2R:z::z: - 2RQR+ R8x - TxR = 0,

m+n

(a + -.!!!:!!:.-ax) T = - 2ma RQ,y m+n n+m

146 Chapter 3

where a is a complex parameter, Q is a rectangular m x n matrix, and R is a rectangular

n x m matrix. The matrices 8 and Tare m x m and n x n matrices respectively.

The system (3.7.8) is equivalent to the condition [L 1 , L 2 ] = 0, where [449]

(_n1 0) (0 Q)n+m m,8x+L1 = 8y+a o '0, -....!ILl Rn+m n

(3.7.9)

( n~mlm, o ) 2 CQ) AL2=8t+a -....!ILl 8x + R o e. +8.0,n+m n

Here, In (1m) is the n x n (m x m) identity matrix, while S is an (n + m) x (n + m)

matrix.

In the case n = m, equation (3.7.8) represents a generalization of the DS equation

(1.3.5) to square matrices. For m = 1 (or n = 1), it is an n(m)-component generalization

of the DS equation.

The DS equation (1.3.5) is the (2 + I)-dimensional analog of the system (1.1.18).

The next system in the hierarchy integrable by the spectral problem (1.1.I7a) , namely,

the system (1.1.19), is rather interesting since it contains the KdV and mKdV equations

as the reductions r = 1 and r = q, respectively. An analogous situation arises in the

(2 + I)-dimensional case.

The (2+ I)-dimensional analog of the system (1.1.19) is

qt+ qeee + qT/T/T/ - 3qe8;;1(rq)e - 3qT/8i1(rq)T/

- 3q8i1(rqT/)T/ - 3q8;;1(rqe)e = 0, (3.7.lOa)

rt + reee + rT/T/T/ - 3re8;;1(rq)e - 3rT/8i1(rq)T/

- 3r8i1(qrT/)T/ - 3r8;;1(qre)e = 0, (3.7.lOb)

where 817 = 8x + (J8y , 8e = 8x - (J8y , and (J2 = ±1. The system (2.7.10) is equivalent

to the commutativity condition for the two operators

where

L1 = (817 0) + (0 q),o 8e r 0

3 n3 (8e 0)L2 = 8t + 8e+ a~ + 3Q + 3P,o 817

(3.7.11a)

(3.7.11b)

(3.7.12)

Other Integrable Equations and Methods in 2+1 Dimensions 147

The system (3.7.10) with (1 = 1 was first introduced in the paper [449] wherein both

the inverse problem and the Hamiltonian structure were studied. The system (3.7.10)

with (1 = i and operators Ll and L2 given by (3.7.11) - (3.7.12) were first presented in

[450]. It was proved therein that this system is Lagrangian.

It is readily seen that the system (3.7.10) admits the reduction q == 1, r = u (or

q = u, r == 1). Under this reduction, equation (3.7.lOa) is satisfied identically and

equation (3.7.lOb) is reduced to the NVN equation (1.3.25) with kl = k2 = 1.

It may also be shown that the system (3.7.10) admits the reduction r = q = v and

r = it = v. The first reduction, subject to the additional constraint v = V, is natural for

the system (3.7.10) with (1 = 1. In this case, that system (3.7.10) reduces to the single

equation [449]

8- 1( 2) 8-1( 2) 3 8-1( 2) 3 8-1( 2)Vt+v~~~+v'7'7'7-3v~ '7 v ~-3v'7 ~ v '7-2V ~ v '7'7-2v '7 v ~~ = O. (3.7.13)

For r = it := v, the system (2.7.10) with (1 = i reduces to [450]

where z = 4(x + iy ), z = 4(x - iy ). It is natural to refer to equations (3.7.14) and

(3.7.13) as the modified NVN-I equation and modified NVN-II equat ion, respectively.

In the (1 + I)-dimensional case, the fact that the KdV and mKdV equat ions arise

as different reductions of the same system (1.1.19) allow us to find immediately the

transformation which relates these two equations. Indeed, under the reduction q ==1, r = u the spectral problem (1.1.17a) is equivalent to the problem (-a~+U) 1/11 = >.21/11,

where 1/; = (1/;11/;2f, while under the reduction r = q := v, one has -8;(1/;1 + 1/;2)+ (Vx +V2)('lfll +'IfI2) = >..2('lfll + 'IfI2). Hence, in both cases, we arrive at the st at ionary Schrodinger

equation. The Miura transformation u = V x + v 2 follows. This simple argument may

be generalized to the N x N matrix problem and gives generalized (1 + I) -dimensional

Miura transformations [209,451] .

An essentially different situation arises in the (2+ 1)-dimensional case. The problem

L 11/1 = 0, where L 1 is of the form (3.7.11a) under the reduction q == 1, r = u , that is,

the 2 x 2 problem

(8'7 0) (1/;1) + (0 1) (1/;1) = 0o 8~ 1/;2 U 0 1/;2

(3.7.15)

is obviously equivalent to the problem (-8~8'7 + U) 1/11 = O. However, the problem

corresponding to the reduction r = it = v , namely,

(8'7 0) ( 1/11) + (0 v) ( 1/11) =0,o 8~ 1/12 V 0 1/12

(3.7.16)

148 Chapter 3

does not, in general, reduce to a scalar problem of the form -Oe0.,.,('l/Jl +'l/J2) +P(V)('l/Jl +'l/J2) = 0. It is, however, possible to achieve this if one imposes some additional constraint

on v. In the case a = 1 and v = V, this condition is ve = v.,." that is, the condition

of the one-dimensionality. Thus, a two-dimensional Miura-type transformation which

relates the mNVN-II equation (3.7.13) and the NVN-II equation is not available.

In the case a = i (a.,., = ai, oe = oz), the constraint, namely, V z = Vi [450] is not

too restrictive [450] . With this condition, (3.7.16) is equivalent to the scalar problem

(3.7.17)

One can also show that the problem L2'l/J = 0, where L2 is given by (3.7.11b) with a = i,

is equivalent to the corresponding problem for the NVN-I equation with u = -vz + VV.

Hence, if the function v obeys the mNVN-I equation (3.7.14) and the constraint V z = Vi,

then the function [450]

u = - vz +vv (3.7.18)

is a solution of the NVN-I equation [450] . The transformation (3.7.18) represents

a Miura transformation between the mNVN-I and NVN-I equations subject to the

restriction mentioned above.

As far as the integration procedure for the mNVN-I and mNVN-II equations with

vanishing asymptotics v ---> °is concerned, one can employ the results of Sectionsx 2+ y 2.....oo

2.4 and 2.5 with the additional restrictions imposed by the reductions r = q or r = ij.

For the inverse problem data, these reductions in the case of the mNVN-I equation are

(see (2.4.39))

¢(k)l1 = </>(k)22' >"k = Jik ' "Yk2 = 1'kl ,

while for the mNVN-II equation

The time-evolution of the inverse problem data for the mNVN equation is given by

formulae of the type F1(>..).,t) = F1(>..).,0)exp [~(>..3 + >,3)t] , and we arrive at the

standard 1ST scheme for the solution of the initial value problems for the mNVN-I and

mNVN-II equations (3.7.14) and (3.7.13). In the case of the mNVN-I equation, we

obtain a pure a-problem and for the mNVN-I equation, a nonlocal Riemann-Hilbert

problem. For the mNVN-I equation with the asymptotics v ---> 1, one has ax 2+ y 2.....oo

combination of a a-problem and a nonlocal conjugation problem on the unit circle

[368] .

Other Integrable Equations and Methods in 2+1 Dimensions 149

We next consider integrable systems which possess a triad operator representation

[L1,L 2] = BL1 with second order operator L1. The first example is [266]

Vt + ~v - a{v2)~ + 2{3V'18i1V'1 - 2a8~lu~~ + 2{3{r1U'1'1 = 0,

Ut - ~u - 2a{uv)~ + 2{3{u8i1V'1)'1 = 0,(3.7.19)

where v{x, y, t) and u{x, y, t) are scalar functions, ~ = a8i - {38~, 8~ = 8:z; - a8y, 8'1

= 8:z; + a8y , a2 = ±1, and a,{3 are arbitrary constants. For the system (3.7.19),

L2 = 8t + a8~ + (38~ + 2{3{8i1V'1)8'1 + 2a8~lu~,

B = 2~8ilV.

(3.7.20)

In terms of the potential q with v : q~, the system (3.7.10) adopts the form (1.3.24).

In the particular case a = -1, (3 = 0, a = 1, the system (3.7.19) reduces to

(3.7.21)Ut + u~~ + 2{uv)~ = O.

The system (3.7.21) is a {2 + 1)-dimensional integrable generalization of the one­

dimensional long dispersive wave equations and was introduced in [267]. Therein, the

system (3.7.21) was constructed via the weak commutativity condition

(3.7.22)

It is readily seen that the weak commutativity condition is equivalent to the triad

operator equations [L1,L2] = BL1, where B is some operator. The inverse problem for

the equation L1'l/J = 0, where L1 is given by (3.2.20) with a = 1, has been studied in

[406].

In the terms of the new variables q = -2v, r = -1 + 4u - 2v'1 the system (3.7.21)adopts the form [406]

1 2qt'1 - r~~ - 2{q )~'1 = 0,

rt~ - (qr + q+ q~'1)~~ = O.

This system is similar in form to that used to describe long waves in the plane in [452].

The system (3.7.21) can, in fact, be rewritten as a single equation either in terms of

U or valone on appropriate elimination. On introduction of the new dependent variable

cp := In 4u, since

150

the equation for u (with ~ -+ x, 'TI -+ t, t -+ y) takes the form (267)

(e-'I' [e'l'{cpxt + sinh cp)xJ) x

+ (e-'I'8;1{e'l')y)yt - 4[(e-'I'8;1(e'l')y)2Lt = o.

Chapter 3

(3.7.23)

This equation is of interest since, in the one-dimensional limit 8ycp = 0 (if ip -+ 0 at

x -+ (0), it reduces to the sinh-Gordon equation CPxt + sinh e = O.

The system (3.7.21) has a solution of the form (267)

1u{~, 'TI, t) = 4(1 + 48e8rP ),

(8t817 - 8l817)Dv(~, 'TI, t) = -2 1+ 48 8 D 'e 17

(3.7.24)

where D{~, 'TI, t) = InJdp{>.) exp [>.~ - 4~ 'TI + >.2t ] , r is an arbitrary contour, and

rdp(>.) is an arbitrary measure. The corresponding solution of equation (3.7.23) is (267)

cp(x, y, t) = In(l + 48x8tD(x, t, y)).

The second example of a system with a triad representation is (266)

qt +!:::..q - a(qe)2 + /3(q17)2 + 2aqePe - 2/38"i 1(qeP17)17 = 0,

Pt - !:::..p+ a(pe)2 - /3(P17)2 + 2/3q17P17 - 2a8;;1(qep17)e = 0,(3.7.25)

where a and /3 are arbitrary constants and zx = a8l-/38~. The corresponding operators

£1, £2, B are given by

£1 = 8e817 + qe 817 + P178e,

£2 = 8t + a8; + /38~ + 2ape8e+ 2/3q17817'

B = 2!:::..{q - p).

The system (3.7.25) contains the independent variables and dependent variables F, 'TI on

a more symmetric basis than the system (3.7.10) or (1.3.24).

The final example is the following matrix system (252)

Vt + Ave - [A,8;;1ud - [A, v)v - [v, [A,8;;1v)) = 0,

Ut + ueA- [A, vu)- [u , [A, 8;;1U )) = 0,(3.7.26)

Other Integrable Equations and Methods in 2+1 Dimensions 151

where u(~, TI, t) and v(~, TI , t) are matrices of arbitrary order and A is a constant matrix.

For the system (3.7.26) [252],

B=[v,A].

It is noted that the system (3.7.19) admits the reduction u = 0, while the system

(3.7.25) admits the reductions p = °or q = 0. Hence, we arrive at the equation

(3.7.27a)

or, with q~ := v,

(3.7.27b)

Equation (3.7.27a) or equation (3.7.27b) are (2 + I)-dimensional analogs of Burgers

equation. The change q = -lnf (or v = -(lnf)~) is the (2 + I)-dimensional analog of

the well-known Hopf-Cole transformation [453, 454] . It is readily verified that, if thefunction f obeys the linear equation

ft + ilf = 0, (3.7.28)

then q = -lnf satisfies equation (317.27). This Hopf-Coletype transformation allowsus

to construct exact solutions of the (2 + I)-dimensional Burgers equation (3.7.27) from

solutions of the linear equation (3.7.28). For a: = -{3 , equation (3.7.27) is invariant

under rotations in the plane and has the form

1 - 2-qt + ilq - ('Vq) = 0,a:

(3.7.29)

(3.7.30)

where il = (,{7)2, aX l == a~, aX 2 == a". Equation (3.7.28) is the two-dimensionalheat equation. Some properties of the (2 + I)-dimensional Burgers equation have been

considered in [455, 456].

Another example of a linearizable system is given by

Ut + a:ilu + a:v.~u = 0,

Vt + 2a:~u - a:[u, V] = 0,

where u(x, t) is a scalar function, v(x, t) is a two-dimensional vector, V is the gradient

operator, il = (~)2 is the Laplace operator, and x = (X1,X2)' The system (3.7.30) may

be represented in the form [L 1, L2] = 0, where

L1 = il + vV + u,(3.7.31)

152 Chapter 3

An interesting feature of the system (3.7.30) is that it is invariant under rotations in

the plane. The system (3.7.30) and the operators (3.7.31) as well as equation (3.7.29)

are evidently generalizable to an arbitrary multidimensional space.

The system (3.7.30) is linearized by the transformation u{x, t) -+ g{x, t) defined by

the formulae [272]u = _g-ll::!.g,

and the corresponding linear equation for 9 is

(3.7.32)

gt + o:l::!.g = O.

To conclude the present discussion of various integrable equations connected with

2 x 2 matrix and second order differential auxiliary linear problems we consider the

interrelation between these spectral problems. It has already been mentioned that the

equation {-8~8'7+u)1/J = 0 may be represented in the 2x2 matrix form (3.7.15). Indeed,

the equation

{8~8'7 + v8'7 + u)1/J = 0

is also representable in a 2 x 2 matrix form, namely,

(8 0) (0 -1)'7 if> + if>=0.o 8~ u v

(3.7.33)

(3.7.34)

The form (3.7.34) of the problem (3.7.33) turns out to be useful in the analysis of

(3.7.33).

It is interesting to note that the standard two-dimensional 2 x 2 matrix problem,

namely,

(8'7 0). ( 0 q) .1/J + t/J = 0,o 8~ r 0

is also equivalent to the problem (3.7.33). Indeed, a simple computation gives

Whence, the problem (3.7.35) is equivalent to (3.7.33) with

(3.7.35)

v = -(lnq)e, u = -rq. (3.7.36)

In particular, the problem (3.7.35) with nontrivial asymptotic values roo, qoo is equiv­alent to

(8e 8'7 + v8'7 + u)1/J = €1/J

with energy € = qooroo.

Other Integrable Equations and Methods in 2+1 Dimensions 153

The correspondence (3.7.36) leads also to the equivalence of equations integrable

by the problems (3.7.35) and (3.7.33). For instance, the system (3.7.21) is equivalent to

qt+ qt;t; - 2qa;;1(qr)t; = 0,(3.7.37)

rt - rt;t; + 2ra;;1(qr)t; = 0

which represents a particular form of the DS equation (1.3.5). Under the reduction r =ii, the system (3.7.37) coincides with equation (1.3.8) up to a change of the independent

variables.

In view of the correspondence (3.7.36), the conservation laws, symmetries, and

Backlund transformations for the systems (3.7.37) and (3.7.21) are equivalent. More­

over, all the formulae which appear in the integration procedures for these systems are

likewise equivalent. In particular, taking solutions of system (3.5.5), we may obtain thesolutions of the system (3.7.21).

The fact that the 2 x 2 matrix systems (3.7.34) and (3.7.35) are matrix forms of

the same scalar problem indicates that they are not independent. Indeed, it may be

shown that the spectral problems (3.7.34) and (3.7.35) may be transformed into each

other via the gauge transformation ,,/J = G¢ ,where

(3.7.38)

In this case, u = - rq and v = -(lnq}t;. Hence, the carryover of the results for thespectral problems (3.7.34) and (3.7.35), one to the other , is achieved via the simplegauge transformation (3.7.38).

One (2+ Ij-dimensional integrable system which stands distinct from the integrableequations considered above is the Benney system [457J

(3.7.39)

The system (3.7.39) describes a long wave approximation to the motion of an incom­

pressible nonviscous fluid bounded by a free surface. Here, x with -00 < x < 00 is

the horizontal coordinate, y ~ 0 is the vertical coordinate, u(x, y, t) is the horizontal

component of the velocity, and h(x,t) denotes the height of the free surface above the

point (x, 0) at time t. The system (3.7.39) may be imbedded in the 1ST method in

a nonstandard way. Let us introduce the moments An(x,t) = fo'1 dyun(x , y, t) . These

obey an infinite system of equations [457J

An,t + An+1,x + nAn-1Ao,x = O. (3.7.40)

154 Chapter 3

(3.7.41)

This system possesses the commutativity representation [L1, L2] = 0 with operators L1and L2 of the form [458]

8 8L1 = (1+ <P>.) 8x - <Px 8>"

8 8L2 = 8t +>.- - Ao -8x ,x 8>"

where <p(>') := Johdy(>. - u(x,y,t))-l = Ef::oAk>.-l-k . Various properties of the

Benney equation (3.7.39), such as, the integrals of motion and the Hamiltonian structure

have been studied in [458-461] .

In conclusion, we note that the main feature of the 1ST method which involves the

representation of a nonlinear equation as a compatibility condition for a linear system

can be generalized to equations with functional derivatives. An example of such an

equation has been given by Polyakov in [462]. This is a three-dimensional Yang-Mills

field in a contour formulation.

This completes the description of (2+ I)-dimensional nonlinear systems integrable

by the 1ST method. It has not been possible to exhaust all (2+ I)-dimensional integrable

equations which may be of interest. Further such equations, notably three-dimensional

difference equations, can be found in [30, 260, 261, 268, 405, 433, 434, 440, 448].

Chapter 4

General Methods for the Construction of

(2+1)-Dimensional Integrable Equations.

T-Function and 8-Dressing Methods

In the two preceding chapters, we have considered a variety of {2+1)-dimensional

nonlinear systems integrable by the 1ST method. The latter procedure relies on the

existence of an appropriate linear system whose compatibility condition is equivalent to

the nonlinear system under consideration . No general criteria for the applicability of

the 1ST method to a nonlinear system are presently available. Accordingly, each specific

case requires a separate investigation.

An effective procedure for the construction of integrable equations and their so­

lutions has been developed by Zakharov and Shabat [24, 26] . The so-called dressing

transformations play a central role in their method . A group-theoretical structure

underlying the dressing transformations for (2+1)-dimensional integrable equations

which may be represented in the form (1.3.1) has been described by Date et al. in a series

of papers [56, 57, 463-469] . It turns out that the theory of integrable equations is closely

connected with infinite-dimensional analogs of the classical Lie algebras An, l3n,en, and

'Dn . The so-called r-function plays a pivotal role in this connection and, in particular,

allows us to represent hierarchies of integrable equations in a very compact form.

Parallel developments in the dressing method have taken place. Following the

discovery of the nonlocal Riemann-Hilbert and a-problem, it was seen that such for­

mulations are related to methods for the construction of multidimensional integrable

systems (Manakov and Zakharov [270, 271]) . Indeed, the construction procedure based

on the nonlocal a-problem (1.4.13) is currently the most general version of the dressing

method. The r-function and a-dressing methods along with the operator representation

of multidimensional integrable equations will be the subject of the present chapter.

155

156 Chapter 4

4.1. The r-function, vertex operator, and infinite-dimensional groups for

the KP hierarchy

The KP equation is the simplest member of an infinite family of equations which

may be represented in the form (1.3.1). The variable x is distinguished in this case

with respect to the variables y and t which arise on a relatively equal footing. For

higher-order equations of the form (1.3.1), the variable x is denoted by Xl, while y and

t are denoted by Xn and X m . The condition (1.3.1) is then written in the form

(4.1.1)

n

where un(x, ax,) = L Uno(X)~,. Each equation (4.1.1) depends only on the variables0=0

A unified treatment of all equations of the type (4.1.1) underlies the r-function

method which is here treated as in the papers [56, 57, 463-469] . The first step consists

in the representation of equations (4.1.1) as a single infinite-dimensional system [463] .

In this formulation, the coefficients Uno of the differential operators Un, Vm depend

on an infinite number of variables Xl, X2, X3, .... The equations (4.1.1) arise as the

compatibility conditions for the infinite linear system

n = 2,3,4, ... (4.1.2)

where a== ax, . When the Uno and Um o depend only on the three variables XI,Xn , andX m , the infinite system (4.1.1) (n,m = 2,3, 4, . . .) yields associated integrable equations

such as the KP equation. The latter arises in the case U20 = U2o(XI ,X2,X3), U30 =

U30 (XI, X2, X3) with the other Uno constant. The KP hierarchy as described in Sections

2.1 - 2.2, corresponds to the hierarchy of reductions U20 = U20(XI ,X2,Xm), Um o =umo(XI,X2,Xm) (m = 3,4,5, .. .). The infinite system of equations (4.1.1) will be

referred to as the generalized KP hierarchy, [56, 57, 463-469].

In what follows, we restrict our attention to systems (4.1.2) of the form

(4.1.3)

where Uno ~ 0 as IXII~ 00. The wave function W(XI, ' •• ) encapsulates information

about the integrable system. For Uno == 0, the system (4.1.3) has the solution tlJo =

exp~(x, >.), where ~(x, >') = Xl>' + X2>.2 + X3>.3 + ... and>. is an arbitrary parameter.

Let us consider the solution W of the system (4.1.3) with Uno ::f 0 of the form

W(X , >') = X(x,>') exp~(x, >.). (4.1.4)

Methods for Construction of (2+1)-Dimensional Integrable Equations 157

The function X(x, >') obeys the system of equations

n = 2,3,4, .. . . (4.1.5)

It is noted that the function X(x,>') and spectral parameter>. are introduced in the

same way as for the KP equation (see Sections 2.1 - 2.2).

A rigorous proof of the existence of solutions of the type (4.1.4) (Baker-Akhiezer

functions) for the system (4.1.3) has been given by Krichever in [348,349]. It may also

be proved that the function X admits the following formal representation

X(x, >') = 1+ Xl (x)>' -1 + X2(>')>' -2 + ....

By virtue of (4.1.6), the solution (4.1.4) has the formal representation

w(x, >') = P(x, 8)ef,(x,>.) ,

were P(x,8) is the operator

P(x,8) = 1+ XI(x)8- 1 + X2(x)8- 2+ ...

(4.1.6)

(4.1.7)

(4.1.8)

and 88- 1 == 1. Operators of the form (4.1.8) are referred to as pseudo-differential

operators. They arise commonly in the theory of nonlinear integrable equations [59, 60,

70, 74] . The theory of such operators is well developed (see, for example, [470]). The

multiplication law is given by

00 n m

= L L L c~(ailFbk)8i+k-j. (4.1.9)j=O i=-oo k=-oo

We shall denote that differential and integral parts of the pseudo-differential operator

Pas P+ and P_:

It is noted that

n

(P(x,8))+ := L ai(x)8i,i=O

-1

(P(x,8))_:= L ai(x)8i.

i=-oo

8ef,(x,>.) = >.ef,(x ,>.) .

(4.1.10)

(4.1.11)

158 Chapter 4

Whence, on application of the operator P(x, 8) of the form (4.1.8) to the left-hand side

and right-hand side of (4.1.11) and on use of (4.1.7), we arrive at the equation

Lw(x, A) = AW(X, A),

where

L(x,8) = P(x, 8)8P- l (x, 8).

The operator L is also a pseudo-differential operator with

L(x,8) = 8 + vo(x) + Vl (x)8- l + v2(x)8-2+ ....

(4.1.12)

(4.1.13)

(4.1.14)

Thus, the solution w of the infinite system (4.1.3) is also the solution of the one­

dimensional pseudo-differential spectral problem (4.1.14) with spectral parameter A

introduced via the formula (4.1.4). Note that er Ie:" = 8 + Vo - 88ep + ..., whereXl

sp is an arbitrary scalar function. Hence, one can always eliminate the coefficient VQ in

(4.1.14) by an appropriate gauge transformation w -+ el{Jw.

The operator L and equation (4.1.12) playa significant role in the description of the

KP hierarchy. The compatibility condition for equation (4.1.12) with equations (4.1.3)

has the form8L-8 = [An ,Ll ,

Xnn= 1,2,3, . .. (4.1.15)

n-l

where An = 8n + E una8a. The operator L generates all the operators An accordinga=O

to

(4.1.16)

Indeed, let L be an operator of the form (4.1.14) with VQ = O. Then , by virtue of

(4.1.15), one has [An - L", Ll+ = O. Let us assume that An - L" = ar (x )8r +..., where

r < n. Then

[A t» Ll Ba;8r 8r - ln - , = - aXl + C + .. . ,

h Ba; 0 R . . f hi d . ld 8ar - l 0 d Cso t at -f) = . epetition 0 t IS proce ure yie s -8-- = an so on. onse-Xl Xl

quently, it is natural to assume that ar = ar-l = . .. = 0, whence we get (4.1.16).

The equations (4.1.1) are consequences of equation (4.1.15) by virtue of (4.1.16).

Since 8aLm = [An, Lml, we obtainXn

Methods for Construction of (2+1)-Dimensional Integrable Equations 159

Hence, the whole KP hierarchy is generated by equations (4.1.15) with the operators

An in the form (4.1.16). This demonstrates how the eigenfunction w(x, -\) encodes

information about the KP hierarchy.

We next introduce equations adjoint to (4.1.12) and (4.1.3), namely,

L*w*(x,-\) = -\w*(x,-\),

( - 8~n + A~(x, 8») w*(x, -\) = 0,where L * and A;' are operators formally adjoint to the operators L and An:

n-l

A~ = (-8t + L(-8)Oun o(x).

0=0

The solution w* of the adjoint equations (4.1.17) and (4.1.18) is of the form

(4.1.17)

(4.1.18)

(4.1.19)

(4.1.20)

where P*(x,8) is an operator formally adjoint to the operator (4.1.8).

Let us now consider the product ¢(x, x', -\) := w(x, A)W* (x', A) of solutions of the

problem (4.1.12) and its adjoint with different points x and x' . This product satisfies

the equationsL(x, 8)¢(x, x', A) = A¢(X,X',A),

(4.1.21)L*(x',a')</J(x,x',>.) = >'</J(x,x',>.)

or(L(x,8) - L*(x',8'Ȣ(x,x',>.) = 0,

(L(x,8) + L *(x', 8'»¢(x, x', >') = 2A¢(X, x', A).(4.1.22)

The properties of ¢(x, x', A) as a function of A are essentially different from the

properties of the functions wand w* in that the residue of the function ¢(x, x', A) at

-\ = 00 is equal to zero. This may be established by appeal to the relation

Res ¢(x,x',-\) = -21. Jd-\W(X,A)W*(X',-\),

A=oo ll"troo

(4.1.23)

where integration is performed along a small contour roo about the point A = 00

and J2dA. = 1. Thus, we first calculate Res¢(x,x',>.) at Xi = x~ (i = 2,3,4, . . .).ll"t A=oo

roo

160 Chapter 4

Substitution of the forms (4.1.7) and (4.1.20) of the solutions wand w* into (4.1.23)

and integration over A yields

~ r!>(x, x', A) 1 1~;';~4' ... = 2~i ! dAX(X, A)eXl~X* (x, A)e-X~xroo

( ,)-l-,-j~ * Xl-Xl

= Li Xj(X)X, (x) (-l-i- .)! .i+j:5-l J

(4.1.24)

Then, taking into account the relation between the functions xi and Xk which follows

from (4.1.20) and (4.1.9), one can show that the right-hand side of (4.1.24) is zero.

Thus, Res r!>(x, x', A)I ...= .. , = o. If we set x~ = Xl + E in this equality and expand in~=oo 11=2,3.4....

E, we obtain

! dA ((8:J Q w(x,A)) W*(X,A) =O.roo

a = 1,2,3, . .. (4.1.25)

Further, since the function w obeys equation (4.1.3) so that 88w = An(x,8)w, fromXn

(4.1.25), one has

! dA((8x l ) QI .. . (8xJ Q nw(x, A))W*(X, A) = 0

roo(4.1.26)

for all nand al, ... ,an' This infinite system of local equalities is equivalent to thesingle bilocal relation [465]

JdAW(X', A)W*(X, A) = 0,

roo(4.1.27)

where x' and X are arbitrary points. Following [465], we shall refer to the fundamental

relation (4.1.27) as the bilocal bilinear identity. One can also establish an inverse result

as in [465] . Thus, ifthe functions W(X ,A) and W*(X',A) given by

00

w(x, A) = L A-nXn(x)e~(x,~) ,

n=O

00

W*(X,A) = LA-nX~(x)e-~(x,~)

n=O

obey the bilinear identity (4.1.27), then they are solutions of equation (4.1.3) and

(4.1.18), that is, W(X,A) is an eigenfunction common to the whole KP hierarchy. Ac­

cordingly, the single bilinear identity (4.1.27) is, in fact, equivalent to the infinite system

of equations (4.1.3) together with their compatibility conditions.

Methods for Construction of (2+1)-Dimensional Integrable Equations 161

In order to extract the information encoded in the bilinear identity (4.1.27) we

introduce a new object, the so-called r-function, This is defined by the relations [463]

n = 1,2,3,. . . (4.1.28)

where the function X is given by (4.1.4) and obeys equation (4.1.5). Another relation,

equivalent to (4.1.28), which connects w(x , A) and the r-function adopts the form [463]

X(A)r(x) = w(x, A)r(x),

where the operator X(A) is defined by

with fJ:= (aXt ' ~aX2' ~aX3' ...). Correspondingly,

X*(A)r(x) = w*(x, A)r(x) ,

where

The relation (4.1.29) can also be rewritten in the form

(4.1.29)

(4.1.30)

(4.1.31)

(4.1.32)

It should be noted that the relation (4.1.29) represents, in fact, a linear problem

in which w(x ,A) can be treated as the potential and rex) as the eigenfunction. In the

original linear equations (4.1.2), the functions uno:(x) are the potentials and w(x, A) is

the eigenfunction. In turn, w(x, A) serves as the potential for the eigenfunction rex) in

equation (4.1.29).

Thus, we have something akin to a pyramid at the base of which resides the infinite

family of equations (4.1.1), (4.1.2), while the r-function is situated at the apex. The

information about the whole KP hierarchy is encoded in t he r-function through a single

equation which results from (4.1.27) and (4.1.32), namely,

JdAe~(x-x").)r(x - f(A-1»r(x' + f(A-1»= 0,

roo

( - 1) (1 1 1 )where fA := A' 2A2 ' 3A3" " .

V x,x' (4.1.33)

162 Chapter 4

By virtue of (4.1.28) and (4.1.16), the mixed derivatives 8x;8x k Inr(x) are differ­

ential polynomials in Uno (x). For instance, 8Xl8x n In r = nUn+l (x) +C2'8xlUn together

with terms containing U2," " Un-I. Hence, the relation (4.1.33) is equivalent to an

infinite family of nonlinear equations for the coefficients unQ(x) of the operators An. In

order to obtain a more transparent description of this infinite family, it is convenient to

represent it in the bilinear form. To this end, we perform in (4.1.33) the change of the

variables x -+ x - y, x' -+ x +Y to obtain

JdAe~(x-X/tA)r(x - f(A -1))r(x' + f(A -1))

roo

= JdAe~(8",A-l)r(x + y)r(x - y)e-2~(y,A)

roo

= JdAL A-i(Pi(8y)r(x + y)r(x - V))L AiPi(-2y) = 0,roo i e

00

where the functions Pi(X) are defined via the expansion exp~(x,A) = LAi~(X) andi=O

A ( 1 1 )By = By l , '2BY2' '3BY3' . .. . As a result, we obtain [465]

00

L Pi(-2y)Pi+l(8y)r(x + y)r(x - y) = 0.i=O

(4.1.34)

This is a single but bilocal (x and yare arbitrary) equation for the r-function, This

bilocal representation is equivalent to an infinite family of local equations. Equation

(4.1.34) can be rewritten in the form

(4.1.35)

If one now adopts the Hirota notation (see (3.2.9»

P(D)f(x).g(x) := P(8y)f(x + y)g(x - y)ly=o'

then it is seen that (465)

(4.1.36)

A 1where D X I := - DX I '

n

Methods for Construction of (2+1)-Dimensional Integrable Equations 163

Expansion of the left-hand side of (4.1.36) in a Taylor series in y produces an

infinite family of local equations in bilinear form. The first two nontrivial equations of

this family are

(4.1.37)

(4.1.38)

Comparison of (4.1.37) and (3.2.10) shows that equation (4.1.37) is nothing but the

KP-I equation in Hirota bilinear form (with x -+ Xl, Y -+ X2, t -+ -iX3)' The relation

U20(X) = 2a~ In T produces the KP equation in the usual form, while the relation (3.2.7)

shows that the variable f of Section 3.2 is but a particular case of a T-function . In

other words, the r-function is a generalization of the variable f appropriate to the

infinite KP hierarchy. Equation (4.1.38) represents a higher order member of the KP

hierarchy. In general, we see that the r-function representation (4.1.36) provides us

with a remarkably compact description of the entire KP hierarchy. However, this is

but one important property of the r-function, Thus, the multi-soliton solutions of the

KP hierarchy also have a conveniently compact form in the terms of the r-function.

The N-soliton solutions of the KP equat ion are given by the formula (3.2.11). The

T-function which generalizes (3.2.11) and gives to the N-soliton solution of the entire

KP hierarchy adopts the form [4641

= 1 + L aie~' + L ajakCjke~;+~" + .. .i j<k

N

=Lr=O

(4.1.39)

C oo _ (Pi - Pj)(qi - qj)oJ - (Pi - qj)(qi - Pj)

with ai, Pi, qi arbitrary real parameters.

A remarkable property of t he N-soliton T-function (4.1.39) is that it is generated

by the single so-called vertex operator X(p, q) given by [4641

(4.1.40)

164 Chapter 4

The simplest r-function for the KP hierarchy is r == 1. Ifwe act on To = 1 with the opera­tor eaX(p,q) , then (4.1.40) showsthat eaX(p,q).1 coincideswith the one-soliton (N = 1) r­

function (4.1.39). Furthermore, a direct calculation gives (X(p, q))2exp(E(~(X,Pi) -i

~(x, qi)) = 0 so that (X(p, q))2r (x ) = 0, whence,

eaX(p,q)r(Xj al,P1, q1j . .. ;aN,PN, qN) = r(xj al,P1, q1; . .. ;aN,PN,qn , a,p, q).

(4.1.41)

Accordingly, the operator eaX(p,q) converts the N-soliton solution into the

(N + 1)-soliton solution.

The action of the operator eaX iterated N times on rn = 1 gives rise to the general

N-soliton -r-funotion (4.1.39) via

r(x· a P q . . a P q) - eI:~l aiX(Pi,qi) 1, 1, 1, 1,···, N, N, N -.- . . (4.1.42)

Thus , the soliton r-functions are obtained by iterated action of the operators eaX(p,q) on

the trivial r-function ro = 1. From (4.1.41), it is seen that the transformations eaX(p,q)

act transitively on the soliton .r-functions. The above properties of the r -function make

it a key notion underlying KP hierarchy theory.

The transformations of the r-function generated by the vertex operator X(P, q)

represent a concrete realization of the general Zakharov-Shabat dressing transformations

(see Sections 1.2 and 1.3). These transformations form an infinite-dimensional system

for which the operators X(P, q)( -00 < p, q < 00) can be regarded as an infinite set ofgenerators. In order to determine the algebraic structure of the transformations of theform eI:i aiX(Pi qi), we must calculate the commutator of the generators. The explicit

form of the operator X(p, q) shows that

[X(p,q), X(p',q')] =

(1- :)_~-~) ,(~) X(p,q') _ (1- ~)_~-~) ,(;) X(P',q), (4.1.43)

(4.1.44)[Z(p,q), Z(p',q')] = 0 (;,) Z(p,q') = 0 (;) Z(p',q).

+00

L1 r 1

t" = -1- + 1 -1· If one sets Z(p, q) := -q-X(p, q), then it is-t -t p-q

n=-oo

where o(t) :=

seen that

Hence, the operators Z(p, q), (-00 < p, q < 00), form an infinite-dimensional

Lie algebra defined by the commutator relations (4.1.43) or (4.1.44). The relations

(4.1.43) or (4.1.44) form a basis for the study of the algebraic properties of the dressing

Methods for Construction of (2+1)-Dimensional Integrable Equations 165

transformations. However, in practice, it proves more convenient to use another infinite­

dimensional basis for this purpose. Thus, let us expand Z{p, q) in the Laurent series

Z(p,q) = LPiq-iZii' Substitution of this expansion into (4.1.44) yieldsi,i

(4.1.45)

where i, i, i',j' are arbitrary integers and O{i) := {~: ~ ~ ~ .

The relations (4.1.45), when the indices i,j,i',j' take a finite number (N) of the

positive values, are nothing but the commutator relations for the classical Lie algebra

AN (or gl(N» (see, for example, [471]). The full infinite-dimensional algebra (4.1.45)

can be considered as the infinite dimensional limit Aoo of the algebra AN as N -> 00

with the inclusion of the unit element 1 (the central extension) [56, 57, 465]. Thus, the

algebra of the dressing transformations of the form eEi aiX(Pi ,qi) is isomorphic to the

infinite-dimensional algebra Aoo with central extension. This result, which establishes

the algebraic structure of the dressing transformations for the KP hierarchy, is one of

the most attractive consequences of the r-function approach .

The interrelation between the KP hierarchy and the algebra Aoo suggests the

possible existence of other infinite hierarchies of equations associated with infinite­

dimensional analogs of the other classical Lie algebras Bn,en, and Dn. In fact, such

hierarchies of integrable equations can indeed be constructed by the same method as the

above KP hierarchy (here termed the AKP hierarchy), and it is natural to refer to these

other hierarchies, in turn, as the BKP, CKP, and DKP hierarchies [56, 57, 466-489] .

Here, we present the basic formulae for the BKP hierarchy [466]. This hierarchy

arises as the compatibility conditions for the system of the form (4.1.3), where n takesonly the odd values 3,5,7, ... . The analog of the bilinear identity (4.1.27) is

Jd>'TW(x, >.)w(x' - >') = O.

roo

The r -function is introduced in analogy to (4.1.31) according to

(4.1.46)

(4.1.47)'TBKP(XI- ~,X3 - 3~3,X5 - 5~5,· . ·)ee(x ,>.) = w(X,>')'TBKP(X),

where €(x, >') = Xl>' + X3>.3 + X5>.5 + .... The infinite hierarchy which follows from

(4.1.46), on use of (4.1.47), is given by the relation [466]

(4.1.48)

{( >') 00' . ' 1 1where e 1/, = E. Pi(y».' and o, := (DX ll -3Dxa, -5Dxs, " .).

l=O(odd)

166

The simplest of the equations (4.1.48) is

Chapter 4

(4.1.49)

Ifwe rewrite equation (4.1.49) in terms of the original variable v = a~ InT, then it is seen1

that it coincides with equation (3.7.1) if we let Xl --+ X, X3 --+ -y, Xs --+ -gt,v --+ 3u.

The CKP and DKP hierarchies are constructed analogously [468]. The simplest

equation of the CKP hierarchy coincides with equation (3.7.3) up to a rescaling of the

independent and dependent variables.

In addition to the basic AKP, BKP, CKP, and DKP hierarchies, one can also

construct the corresponding l -l'-modified hierarchies [57] . For the A case, the l - l'­

modified hierarchy is defined by the following bilinear identity [57] :

JdAAl - l ' e{(X-X',A)Ti(X - f(A -1 ))Tl' (X' + f(A -1)) = 0,

roo(4.1.50)

where l -l' ~ O. The simplest examples of the modified AKP equations with l -l' = 1

are

(4.1.51)

(4.1.52)

In terms of the original variables v = In Ti+1 , equation (4.1.52) becomes the mKP-IITl

equation (3.6.1) with the changes t --+ -~X3'X --+ X1,Y --+ -X2 ,Vx --+ 2v, while the

relation (4.1.51) is nothing but the two-dimensional Miura transformation (3.6.4).

Numerous other explicit examples both ofthe basic and l-l'-modified KP equations

for each of the A, B, C, and D cases can be found in [57, 466, 468]. In each case, one

can introduce analogs of the vertex operator (4.1.40) and establish formulae similar to

(4.1.41) and (4.1.42). The algebras Boo,Coo, and Doo arise instead of algebra (4.1.45)

and correspond to special types of spectral problem (4.1.2).

The preceding constructions possess an interesting algebraic and field theory re­

alization. This is based on an infinite-dimensional generalization of the well-known

realization of classical Lie algebras in terms of the creation and annihilation operators

of bosonic and fermionic types (see, for example, [472]). In the Aoo , Boo , Coo , and

Doo cases, one starts with an infinite-dimensional Clifford algebra A with generators

1/Ji,1/J;(i E Z) which satisfy t he defining relat ions

(i ,j E Z) (4.1.53)

Methods for Construction of (2+1)-Dimensional Integrable Equations 167

where [7/J , tp]+ := 7/Jtp + tp7/J. The generators 7/Ji' 7/J; can be t reated as classical anticom­

muting fermion fields. The infinite-dimensional algebras Aoo, Boo, Coo, Doo are then the

subalgebras of the enveloping algebra E(A) generated by the quadratic combinations of

the elements 7/Ji' 7/Jic. Indeed, one can easily check that

(4.1.54)

Hence, 7/Ji7/J; (i, j E Z) together with unity generates the infinite-dimensional algebra

Aoo . Consequently the theory of the algebra Aoo can be imbedded in that of the Clifford

algebra (4.1.53). In particular, the representation theory of the algebra Aoo is also

imbedded in the representation theory of the algebra (4.1.53). The infinite-dimensional

representations of the Clifford algebra can be constructed in a standard field-theoretic

manner. The simplest nontrivial representation is the so-called Fock representation. The

various results of this section have a very elegant interpretation within the framework

of the free fermion field realization [56, 57]. For instance, the r-function turns out to be

the character of the representation of the algebra Aoo . We recommend the interested

reader to the papers [56, 57, 463-469].

The r-function has also been introduced in the context of holonomic quantum field

theory (see, for example, [473-477]). The concept has been generalized in [475-477]. The

role of the r-function in the theory of the KP equation was revealed by M. Sato and Y.

Sato [478, 479J (see also [486]). A series of papers [463-469J and reviews [56, 57J appeared

subsequently and t he r-function now occupies an important place in 1ST t heory (see, for

example, [72, 107, 481-48]). The generalization of the above constructions to the case

ofthe supersymmetric KP hierarchy has been given in [499, 500]. The relation between

the r -function and the Fredholm determinants for the KP hierarchy was considered in

[501-503] . Finally, points of contact between KP hierarchy theory and string theory

have been recently uncovered [358, 504-506].

4.2. A generalization of the dressing method

The original and simplest version of the dressing method as proposed in [24] admits

a generalization to more complicated two-dimensional spectral problems, notably to

integrable equations which possess a triad representation [1.3.21]. Here, following the

work of Zakharov in [252, 261J , we show how the dressing method can be generalized

to nonlinear equations which arise as the compatibility conditions for auxiliary linear

systems of the typeL l 7/J := (A1(8x)8y + B1(8x»)7/J = 0,

L27/J := (A2(8x)8t + B2(8x))7/J = 0,(4.2.1)

where Ai(8x) and Bi(8x) are differential operators of the form E:=o Uin(X ,y,t)a;:.

As already seen in Section 1.3, the main distinction between this case and that with

168 Chapter 4

Al = A2 == 1 resides in the fact that the associated integrable equations do not arise

out of the commutativity condition [L1, L2] = 0, but rather, from the condition (1.3.21)

or, indeed, a more complicated operator equation.

It is recalled that the starting point of the dressing method in the case Al = A2 == 1

is the integral operator F which commutes with the undressed operator Lo, so that,

[Lo,F] = O. In our case, the latter condition is too restrictive . For instance, for the

operator Lo = OxOy, the condition [oxOy, F] = 0 gives

(ox + ox,)F(x,x',y) = 0, oxoyF(x,x',y) = O.

Therefore, F(x,x',y) = g(x - x') + fey) so that the problem is one-dimensional.

Accordingly, one must weaken the conditions which should be imposed on the integral

operator F.

Let Lo be a differential operator of the form Lo = Ao(ox)oy + Bo(ox), where

Ao(ox) and Bo(ox) are differential operators in x. In the simplest case, Ao and Bo are

differential operators with constant coefficients. Then, let F be the one-dimensional

Fredholm integral operator

1+ 00

(F1/J) (x, y) = -00 dx'F(x ,x', y)1/J(x', y).

Further, assume that this operator obeys the condition

LoF = GLo,

where G is the one-dimensional Fredholm integral operator

1+ 00

(G1/J) (x, y) = -00 dx'G(x ,x' , y)1/J(x' , y).

(4.2.2)

The operator equation (4.2.2) is equivalent to the following system of differential equa­

tions for the kernels F(x,x',y) and G(x,x',y) :

ofAo(ox) oy + Bo(ox)F - GBd"(ox') = 0,

Ao(ox)F = GAd"(ox')'

(4.2.3)

where At and Bd" are operators formally adjoint to the differential operators Ao and

Bo· The operators At and Bd" act from right to left. Note that, by virtue of (4.2.2),

the operator F transforms any solution 1/Jo of the equation Lo1/Jo = 0 into a solution

F1/Jo of the same equation.

Methods for Construction of (2+1)-Dimensional Integrable Equations 169

Let the operators F and c admit the triangular factorizations

1 + F = (1 + k+)-l(l + k - ),

1 + o= (1 + Q+)- l(l + Q-),

where k± and Q± are Volterra-type operators. From (4.2.2), one has

and, in the sequel, we set

(4.2.4)

(4.2.5)

(4.2.6)

(4.2.7)

It may be verified that the differential operator L is of the form L = A(ox)8y + B(ox).

The formulae which express the coefficients ofthe differential operators A(ox) and B(ox)

in terms of the coefficients of the operators Ao(ox) and Bo(ox) , namely, the dressing

formulae, follow directly from (4.2.7). For example, with Ao = Ox ,Bo = 0, one has

B= oK+(x , x ,y)

oy . (4.2.8)

The dressing procedure is the same as in the case A == 1. Firstly, from equation

(4.2.7) or the system (4.2.3), one finds the kernels F(x ,x',y) and G (x ,x' ,y) . Then, the

kernels K+ and Q+ can be calculated from the Gelfand-Levitan-Marchenko equat ions

F(x ,x',y) + K+(x ,x',y) +l CXJ

dsK+(x,s,y)F(z , x' ,y) = 0,

x'> x

G(x,x' ,y) + Q+(x,x',y) +l CXJ

dsQ+( x , s,y)G(s ,x' ,y) = 0

x' > x

(4.2.9)

which follow from (4.2.4) and (4.2.5). Finally, on use of dressing formulae of the type

(4.2.8) we can calculate the dressed operator L. The dressed function 'l/J = (1 + k+) 'l/Joobeys the equation

(4.2.10)

The main feature of the dressing formulae (4.2.7) in comparison with the case A == 1 is

that it is not now a similarity transformation. This has important consequences.

Let us consider the pair of operators LlO and L20 given by

(4.2.11)

170 Chapter 4

where the coefficients of the operators Ai and B, are functions of x ,Y, and t. Let the

operators LlO and L20 obey equations of the type (4.2.2) so that

(4.2.12)

where 61 and 62 are appropriate integral operators.

The system (4.2.12) is a generalization of the system (1.2.9). The condition which

guaranteed the compatibility of the system (1.2.9) was commutativity of the operators

LlO and L20. In our case, the condition [LlO' L20] = 0 also guarantees the compatibility

of the system (4.2.12). It is seen that LI062L20 = L2061L IO • The undressed operators

LlO and L20 commute, in particular, when their coefficients are constant commuting

matrices. The commutativity of the operators LlO and L20 guarantees also the compat­

ibility of the systemL lQ'lPO = 0,

(4.2.13)

A more general compatibility condition for the system (4.2.13) and simultaneously

for the operator system (4.2.12) adopts the form

(4.2.14)

where N IO and N20 are differential operators of the type

(4.2.15)

The particular case of commuting operators LlO and L20 corresponds to NIO = L20 and

NlO = -LlO.

Use of the formula (4.2.7) shows that the dressed operators

t., = A1(ox)8y + B1(Ox),

L2 = A2(ox)ot + B2(Ox)

also obey an operator equation of the type (4.2.14), namely ,

whereN1= (1 + k+)NlO(1 + Qt)-1 = Cl (ox)Ot + D1(ox),

N2 = (1+ k+)N2o(1 + Qt)-1 = C2(ox)Oy + D2(ox).

(4.2.16)

(4.2.17)

Methods for Construction of (2+1)-Dimensional Integrable Equations 171

The dressed function 'l/J = (1 + k+)'l/Jo satisfies the system of equations

and equation (4.2.16) represents the operator form of the compatibility condition for

this system. In terms of the ordinary differential operators Ai (8J;) , B i(8x ) , Gi(8x ) , and

Di(8x ) , the operator equation (4.2.16) represents the following system [261]:

8A lc.at + DIAl + G2B2 = 0,

8A2G2 8y + D2A2 + GlB! = 0,

8Bl 8B2Gl at + G2 8y + DlB! + D2B2 = O.

(4.2.18)

This is equivalent to the system of nonlinear differential equations for the coefficient

functions of the operators Ai and Bi. Indeed, solving the first equation (4.2.18), we find

Gl and G2. Then, from the second and third equations (4.2.18), we obtain Dl and D2.

Consequently the last equation (4.2.18) represents a closed system of equat ions for the

coefficient functions of the operators £! and £2.

The dressing formulae (4.2.8) automatically give solutions of the nonlinear system.

The simplest dressing corresponds to the choice of the coefficient functions for the

operators Aio(8x ) and BiO(8x ) (i = 1,2) as commuting constant matrices.

Thus, the principal aspects of the classical dressing method as outlined in Section

1.2 may be generalized to nonlinear equations associated with auxiliary linear systems

of the type (4.2.1), with triad or quartet operator representations. In particular, the

scheme described in this section is clearly applicable to the equations considered in

Sections 2.6 - 2.8 since, for these equations, £1 = (oe + v)oTJ + U .

It is noted that the operator condition (4.2.16) can be rewritten in the form

(4.2.19)

(4.2.20)

The system (1.3.26) is an example of an integrable system possessing the quartet

operator representation (4.2.19) .

172 Chapter 4

The operator equation (4.2.19) represents a generalization of the triad represen­

tation (1.3.21) . It reduces to the triad representation in the case A2 == 1. Indeed,

for A2 == 1, from (4.2.18), it follows that C2 = Al,Cl = -1,D2 = B l, and, as a

consequence, ')'1 = O. Hence, the system (4.2.18) reduces to the form

8A l7ft - DIAl - AlB2 = 0,

8B2 8BlAl 8y - 7ft + DlBl + BlB2 = O.

(4.2.21)

The system (4.2.1) contains the variables y and t symmetrically. Hence, obviously

there is another triad representation for Al == 1 (-rl = 0).

Finally, in the case Al = A2 == 1, it follows from (4.2.18) that Ct = -C2 = -1

and Dl = -B2, D2 = -Bl, that is, ')'1 = ')'2 = O. Accordingly, we arrive at the

commutativity condition [L l , L 2] = O.

One important feature of equation (4.2.19) is that the so-called structure constants

')'1 and ')'2 are not invariant under the dressing procedure. Thus, the dressing transforms

the closed system of two operators Ll and L2 into another closed system but with

different ')'1 and ')'2 .

In conclusion, it is remarked that, in the case when the coefficient functions of the

operators Ai and B, (i = 1,2) are independent of x, following a Fourier transform in x,the system (4.2.1) is converted into a one-dimensional system of the type (1.2.4) with

the transposition x +-+ y.

4.3. The general a-dressing method

The generalization of the dressing method discussed in the previous section con­

cerned nonlinear equations with one distinguished variable x. The development of the

dressing method in the (1 + I)-dimensional case when both independent variables are

on an equal footing requires transition to a local Riemann-Hilbert problem. In 2 + 1

dimensions, the natural generalizations of the local Riemann-Hilbert problem involve

nonlocal Riemann-Hilbert and a-problems [270, 271].

The most general version of the dressing method requires one to start with the

nonlocal a-problem (1.4.13). Here, we describe this a-dressing formalism following, in

the main, the papers of Zakharov and Manakov [270, 271].

We consider the nonlocal a-problem

8X(A,X) - II' -, -, ,-, -8X = (x *R)(A, A) := dA 1\ dA X().., A)R(A ,A j A, A) , (4.3.1)

Methods for Construction of (2+1)-Dimensional Integrable Equations 173

where X and R are matrix-valued functions. In order to make the problem (4.3.1)

uniquely solvable, it is necessary to fix the value of the function X(A, X) at some specified

point AO, that is, one must normalize the a-problem. The canonical normalization

involves taking X(oo) = 1. Other normalizations of the type X(A, X) - g(A,X), whereA-+OO

g(A,X) is some appropriate function, are possible. For the local Riemann-Hilbert prob-

lem (1.2.12), the solutions corresponding to two such normalizations with different gl (A)

and g2(A) are connected by the gauge transformation X1(A) -4 XHA) = ::~~~X1(A) so

that they are gauge equivalent. However, a different situation arises for the nonlocal a·problem (4.3.1) in that the normalizations X(A, X) - g(A,X) with different g(A,X) are,

A-+OO

in general, essentially distinct since they are not connected by a gauge transformation.

The existence of different normalizations of the problem (4.3.1) allows us, as will be

seen, to establish deep interconnections between different integrable nonlinear equations

constructed with fixed canonical normalization.

In this section, we consider the case with canonical normalization X(00) = 1 in

detail. It will be assumed that the a-problem (4.3.1) with the canonical normalization

is uniquely solvable. The a-equation is equivalent to the singular Fredholm integral

equation (1.4.14). Hence, the unique solvability of the problem (4.3.1) is guaranteed, at

least, for R(A', X'; A, X) small in norm. The unique solvability of equation (4.3.1) has one

obvious but important consequence, namely, that if the function XO(A) is the solution

ofthe problem (4.3.1) and Xo(>.) -40 as A -4 00, then Xo is identically zero. This result

will be appealed to in the sequel.

New variables Xl, X2, X3 are now introduced into the problem. If it is assumed

that the kernel R in the right-hand side of (4.3.1) depends on these variables, then so

does the function X. The 1ST method requires that the dependence of the function

R(A', X'; A,X; Xll X2, X3) on the variables Xl, X2 , X3 should be defined by linear equations

which allow integrability. Accordingly, let the function R(A',X';A, X; x) depend on the

variables Xl, X2, X3 in the following linear fashion [270, 271)

8R(A',;'; A, Xi x) = t, (A')R(A').' ;A, X; x) - R(A', X';A, X; x)Ii(A), (i = 1,2,3) (4.3.2)Xi

where Ii(A) are matrix-valued rational functions which commute in pairs.

We introduce the so-called long derivatives D, defined by

8j(X,A)Dd(A,X) := 8 + j(x, A)Ii(A).

Xi(4.3.3)

It is clear that [Di , Dk) = 0 (i, k = 1,2,3). The operators D, allow the conditions (4.3.2)

to be represented in the compact form

where Rj := j *R.

i = 1,2,3 (4.3.4)

174 Chapter 4

Application of the operator Di to the left-hand side and right-hand side of the

a-equation (4.3.1) and use of (4.3.4) gives

while, in view of the condition [Di , Dkl = 0, it can be verified that

axM ---=- = M X*R,a>.

where M are differential operators of the form

M = L Uili2i3{XltX2 ,X3)D~1 D;2 D;3

it ,i2ti3

(4.3.5)

(4.3.6)

(4.3.7)

with the Ui1.i2.i3{X) are independent of >.. The operators M form a ring of differential

operators generated by D 1, D2' and D3• The functions MX, in addition to the singular­

ities of the function X, also contain the singularities corresponding to the poles of the

functions li{>').

For the generic operator M, the function MX is not a solution of the a-equation

(4.3.1) as follows from (4.3.6). However, it is a solution of the a-problem if the condition

[~ , M]=O (4.3.8)

is fulfilled. The condition (4.3.8) shows that the operators D, which generate the

operator M have poles at the same points as the functions l iP..). On the other hand,:x >. ~ >'0 = 1l'6{>.->.0). Accordingly, the condition (4.3.8) is fulfilled for those operators

M which have no poles.

Let us assume, for the moment, that operators which obey the condition (4.3.8)

exist . For such operators one has

(4.3.9)

and the solution MX of this a-equation has the same singularities as the solution

X. Hence, by virtue of the unique solvability of the a-problem (4.3.1), Mx{>') is

proportional to X{>'), that is, MX{>') = u{x)X{>') or (M - u{x»X{>') = 0, where u{x) is

a function only of XltX2,X3 . Thus, the ring of operators M which obey the condition

(4.3.8) contains the subring of operators L such that

LX=O. (4.3.10)

Methods for Construction of (2+1)-Dimensional Integrable Equations 175

It is natural to choose a basis in the subring of such operators, L. Let this basis be

L1, • • • ,Ln' For these operators, we have the system of compatible linear equations

i = 1, . .. .n. (4.3.11)

The system (4.3.11) automatically has the common solution X which is the solution of

the original a-problem (4.3.1) and the compatible system (4.3.2).

The A-dependence in operators L of the type (4.3.7) can be removed. To do this,

we introduce

(4.3.12)

into the system (4.3.11). It is easy to see that such a transformation converts the

long derivatives Di] in (4.3.3) into the usual derivatives ox.!. In terms of 'l/J, equation

(4.3.11) adopts the form [270, 2711

Li(oXk)'l/J = L U~~~i2,i3(X1,X2,X3)O~110~;O~;'l/J = o.it ,i2 ,i3

(4.3.13)

Hence, we get a system of linear differential equations which are independent of the

parameter A. Like (4.3.11), the system (4.3.13) is automatically compatible and leads

to a system of nonlinear differential equations for the coefficients Ut~i2,i3 (x). Thus,starting with the a-problem (4.3.1), we arrive via (4.3.2) at a compatible linear system

(4.3.13) with associated nonlinear integrable equations. The existence of operators

which obey the condition (4.3.8) is crucial to this construction. Let us now show that

such operators do indeed exist [270, 271).

We start with the simplest example. Let the quantities II (A),h (A) and 13(A) be

of the form

with

(4.3.14)

where x = xl, y = X2, t = X3. The operators D1, D2, D2 in (4.3.14) have singularitiesof the first, second, and third-order, respectively, at A = 00. For our purpose, we

must construct operators L without singularities. Let us consider the quantity D2X =(Oy + A2)X(X, y, t, A), where X satisfies the a-equation (4.3.1). This quantity has a pole

of the second order at A= 00 since X -+ 1 as A -+ 00. There is also a pole of the second

order at A -+ 00 in the quantity Dh = (o~ + 2AOx + A2)X. The coefficients of A2 in

these expressions coincide so that the quantity

(D2 - D~)x = (Oy - 0; - 2AOx)X(A) (4.3.15)

has no poles of second order at A -+ 00. It has, nevertheless, a pole of first order at

A -+ 00. In order to eliminate this pole, the above procedure suggests that one should

176 Chapter 4

subtract from (4.3.15) another appropriate quantity which likewise has a pole of first

order at >. - 00. The only candidate is DIX. However, the coefficient of>. in (4.3.15)

is -2axX, while the corresponding coeficient in DIX is simply X. This problem can

be removed by introduction of the function v(x, y, t) such that v(x, y, t)X(>.) IA=OO =

-2axX(x, y, t)IA=OO' Then, on subtraction of v(x , y, t)DIX from (4.3.15), we obtain

the expression (D2 - D~ - VDI)X which has no singularity at >. - 00. The operator

D2 - D~ - VDI is just the operator M which we seek. It obeys the condition (4.3.8) and

(D2 - D~ - vDdx satisfies the a-equation (4.3.1). By virtue of the unique solvability

of the a-problem, one has (D2 - D~ - vDdx(>') = u(x, y, t)X(>'), where u(x, y , t) is a

function only of x, y, and t. Hence, we arrive at an equation of the type (4.3.11), namely,

(D2 - D~ - v(x, y, t)D1 - u(x, y, t))X(x, y, t, >.) = o.

Finally, on introduction of the function

'I/J(x, y , t , >.) := X(x, y, t , >.)eAX+A2y,

we obtain the usual linear problem

(ay - a; - v(x ,y, t)ax - u(x, y , t))'I/J = o.

(4.3.16)

(4.3.17)

(4.3.18)

One can construct spectral problems of the type (4.3.13) which contain at in exactly

the same manner. In this case, we start with the quantity D3 X which has a pole of t hird

order at >. - 00. This pole, in turn, can be removed, for instance, by subtracting D~X

from D3X. The resultant quantity (D3 - D~)X has a pole only of the second order at

>. - 00. This pole can in turn be removed by subtraction of the quantity VI (x, y, t)D?X,

where VI(x, y , t) is the function of x ,y , and t , such that, 30x X(x, y, t, >.)IA=OO = VI(x , y, t)

X(x, y, t, >'))A=oo . The resultant expression (D3 - D~ - vID?)X(>') contains only a linear

term in >.. This may be removed by subtraction of the quantity V2(X, y , t)DIX, where

V2(X, y, t) is an appropriate function of x, y, and t. The resulting expression (D3 ­

D~ - vlD? - V2DI)X no longer has a singularity at >. - 00 and, by virtue of the

unique solvability of the a-problem, it is equal to V3(X,y, t)X, where V3(X, y , t) is some

function. Accordingly, we obtain an equation of the form (4.3.11), and following the

transformation X -'I/J = xexp(>.x+ >.2 y + >.3t), we arrive at the linear problem

(4.3.19)

Thus , we have obtained a system of two linear equations, namely, (4.3.18) and

(4.3.19) . This system is automatically compat ible. It is readily seen that the system

(4.3.18)-(4.3.19) is nothing but the system (3.6.5) up to a rescaling of the independent

Methods for Construction of {2+1)-Dimensional Integrable Equations 177

variables. The compatibility condition for the system (3.6.5) is equivalent to the non­

linear system (3.6.6). Hence, starting with (4.3.14), we have derived the auxiliary linearsystem (4.3.18)-{4.3.19) associated with the nonlinear integrable system (3.6.6). The

KP equation is the particular case of this system with v == O.

It is not difficult to show that different linear problems, equivalent to (4.3.14), canbe likewise constructed. This applies, for instance, to the problem

or

{8t - 8y8;,: - W1 {x, y, t)8; - W2{X, y, t)8;,: - W3{X, y, t))t/J = O.

It is also noted that equation (4.3.16) with v = 0, namely,

{8y - 8; - 2>.8;,: - u{x, y, t))X = 0,

(4.3.20)

coincides (up to a rescaling of the independent variables) with equation (1.4.4a) whicharises in KP theory. This follows the introduction of the spectral parameter>. andthe function X via the formula (4.3.17). Hence, the procedure associated with theintroduction of a spectral parameter as discussed in Section 1.4 has a deep interpretationwithin the framework of the a-dressing method.

The example (4.3.14) considered above is but the simplest case of a more generalsituation with

(4.3.21)

where 12{>') and 13{>') are polynomials of order nand m, respectively. However,the overall method of construction of the linear equations (4.3.13) proceeds as for(4.3.14). First, we consider the quantity D2X. This has a pole of order nat>' -+ 00.

This pole may be eliminated by subtraction of the term un{x, y, t)DrX. The poleof order n - 1 which remains in {D2 - unDrh is removed by subtraction of theterm Un - 1{x, y, t)Di-1x, where Un-l (z, y, t) is an appropriate function. Repetition

n

of this subtraction procedure produces the operator M; = D2 - L Uk{X, y, t)Df whichk=l

has no singularity at >. -+ 00. Similarly, one can construct the operator M2 = D3 -m

L Vk{X, y , t)Df which is also without singularities at >. -+ 00. These operators satisfyk=1the condition (4.3.8). Consequently, we arrive at a compatible system of two equationsof the form (4.3.13). The corresponding operators £1 and £2 are nothing but theoperators n

£1 = 8y - L Uk(X, u. t)a;,k=1

encountered previously.

m

£2 = 8t - L Vk{X , y, t)a;k=O

(4.3.22)

178 Chapter 4

(4.3.23)

The example (4.3.21) corresponds to a case with the one distinguished variable

z , We have seen in Section 4.3 that, when one variable is marked out, more general

spectral problems of the type (4.2.1) can arise. We now demonstrate that these can alsobe obtained within the a-dressing formulation [270, 271]. For this purpose, we consider

the case

I - A [. - B1(A) I = B2 (A)1 -, 2 - A

1(A) , a A

2(A) ,

where Ai(A) and Bi(A) (i = 1,2) are polynomials in A, that is, 12(A) and la(A) are

general rational functions. Let us start with the quantity D2X. This has poles at the

zeros of the polynomial A1(A) which is taken to be of degree n. Let us act from the leftn

on D2X with the operator A1(D1) = L Uk(X, y, t)Df, where Uk(X, y, t) are functions ofk=l

X, y, t chosen such that the expression A1(D1)D2Xhas no poles at the zeros of A1(A). Asa result, A1(D1)D2X is a polynomial in A of some degree m so that it has singularities

at A -+ 00. These singularities can be removed by subtraction from A1(DdD2X of them

quantity B1(D1)X = L Vk(X, y , t)Dfx with suitable functions Vk(X, u, t). An operatork=l

M1 = A1(D1)D2 - B1(D1) results which has no singularities and obeys the condition

(4.3.8). Finally, we arrive at the linear problem

The second equation (4.2.1)

may be derived in an analogous manner. We conclude that the version of the dressing

method as considered in Section 4.2 is, in fact, imbedded in the a-dressing method.

We next consider a case which is essentially different from those previously dis­

cussed. Let

i = 1,2,3 (4.3.24)

where Ai are commuting N x N matrices: [Ai,Akl = 0 (i, k = 1,2,3), and all the

complex numbers Ai are distinct: Ai i: Ak (i, k = 1,2,3) .

In contrast to the previous cases, all the independent variables Xli X2, xa are now

contained in the problem on an equal footing . We again consider the quantities D,X.These have additional poles at the points Ai. It is clear that it is not possible to

construct a combination without singularities at the points A1, A2, Aa by using only the

quantities DiX. Hence, it becomes necessary to consider operators of the second order.

The expression DiDkX (i i: k) has additional poles at the points Ai and Ak. These

(4.3.25)

Methods for Construction of (2+1)-Dimensional Integrable Equations 179

poles can be removed by addition of the terms U:kDkX and u~kDiX to DiDkX. The

requirement that the residues at the poles >'i and >'k vanish in the quantity

gives

i (0 XkAi ) -1Uik = - x;Xk + >'k _ >'i Xk'

where Xi:= x(x,>')I>.=>.;. It should be noted that there is no summation over repeated

indices in these and subsequent formulae.

The operators Mik satisfy the condition (4.3.8). The unique solvability of the 8­equation gives MikX = WikX, where Wik is some matrix-valued function of X1,X2,X3.

Since X --+ 1 as >. --+ 00, it follows that Wik = O. Hence,

(4.3.26)

Following the transformation (4.3.12), we arrive at a system of three linear problems,

namely [270, 271],

The linear system (4.3.27) is solvable by construction. The formal compatibility condi­

tions for the system involving the equality of the mixed third order derivatives OX;OXkOXt

1/J (i -I k -I £ -I i) are equivalent to the nonlinear system (1.3.30), that is,

(4.3.28)

The nonlinear integrable system associated with the auxiliary linear system (4.3.27)

can be constructed in another manner by direct use of (4.3.26). Indeed, if we set

>. = >'t (£= 1,2,3) in (4.3.26), we obtain [270, 271]

Ak A- A-Akox;OxkXl(X) + ox;Xt At _ >'k + OXkXl At -' Ai + Xl (At - Ai)(At - Ak)

(4.3.29)

(i, k,e= 1,2 ,3; i -I k, k -I e, e-I i)

180

The equivalence of (4.3.28) and (4.3.29) is readily verified from (4.3.25).

Chapter 4

The above examples demonstrate explicitly how operators may be constructed

which obey the condition (4.3.8) and the linear problems (4.3.13). Their construction

becomes more complicated for increased complexity in the functions I, ().). The proof

of the existence of operators M which satisfy the condition (4.3.8) for generic rational

functions Ii{).) is somewhat lengthy and is omitted here. The details are given in [2711 .

All the two-dimensional spectral problems discussed in previous chapters can, in

fact , be constructed within the framework of the a-dressing method. The method

under discussion is conceptually equivalent to the dressing method. Indeed, for R = 0,

one has X = 1, and the operators L, of the form (4.3.19) are differential operators

with constant coefficients. The formulae which express the coefficients of the dressed

operators L, via the solutions X of the a-problem (4.3.1) can be referred to as the

dressing formulae. In the case (4.3.27), the undressed operators are L~Z) = {)X;{)Xk and

the dressing formulae are given by the relations (4.3.25). The construction of concrete

solutions of the associated nonlinear integrable equations follows the standard dressing

scheme. Thus, given the function R(N, :X';).,};;Xl, X2 , X3) , one calculates the function

X{x,).) via the integral equation (1.4.4) and then calculates solutions of the integrable

equation via dressing formulae of the type (4.3.25). The a-dressing method is currently

the most general method of construction of multidimensional compatible linear problems

and their associated nonlinear integrable equations . It likewiseprovides one of the most

powerful available methods for the construction of exact solutions of such nonlinear

equations.

The use of the nonlocal a-problem (4.3.1) allows us to construct wide classes of

exact solutions to nonlinear integrable equations. In particular, solutions involving

functional parameters are generated by factorized kernels R of the form

n

R()", :X';).,:Xjx) = eF {),' ) L h().', :X')Uk()., :X)e-F {),) ,

k=l

(4.3.30)

3

where fk and Uk are matrix-valued functions and F{)') := LIi().)xi. Substitution ofi=l

(4.3.30) into (4.3.1) yields

ox().. , :x) =~ h () (\ \) -F{A){).. LJ k X Uk 1\,1\ e ,

k=l

where

(4.3.31)

(4.3.32)

Methods for Construction of (2+1)-Dimensional Integrable Equations 181

Use of the Cauchy formula

( -) 1 II ' -, 1 aX(>",x') 1 I d>.' (' -')X x, >. = 21l"i d>' /\ d>' >" _ >. a>.' + 21l"i >" _ >. x >. ,>. ,

e 8e

where C is the entire complex plane, gives

The quantities hk are calculated from the algebraic system

(4.3.33)

(4.3.34)

n

hi + L hkAkl = ~l ,k=l

(f = 1, ... , n) (4.3.35)

where

and

~l:= IJd>' /\ d>'eF(>.)»o; >.)

o

Akt = 2~i IJd>./\ d>' JJd~~ ~;' 9k(>" ,x')e-F(>")eF(>. )ft(>., >.), (k , f = 1, ... ,n).

e e(4.3.36)

The system (4.3.35) arises from (4.3.34) following multiplication by eF (>')uo; >.) and

integration over >..

The solutions of the corresponding integrable equation are expressed via X(>" ,>.) at

the poles of h(>"). Hence, the formulae (4.3.34) - (4.3.36) give a solution which contains2n arbitrary functions of the variables. Indeed, one can construct exact solutions of

all the integrable equations mentioned above in this manner. In particular, in the

case Ik = >..k (k = 1,2,3), (the KP-equation) formulae (4.3.35)-(4.3.36) reduce to the

formulae (2.3.22)-(2.3.24).

The choice of the functions 9k according to

produces solutions for which the function X has the poles

(4.3.37)

Here, the matrix F(>") is assumed to be diagonal and there is summat ion over the

repeated index 'Y. It is noted that each matrix element of the matrix-valued function X

given by (4.3.37) has its own set of poles.

182 Chapter 4

Rational solutions can be likewise constructed. In the scalar case, these solutions

correspond to a kernel of the form

n

R(A', X'; A,Xj x) = ; .eF(),') L 6(A' - Ak)6(X' - Xk)6(A - Ak)6(X - Ak)Sk(A',A)e-F(),)

l k=l(4.3.38)

3

where F(A) = L I.(A)x., Sk(A', A) are appropriate functions, and {A1,'''' A~} com­.=1

prise a set of isolated points distinct from the poles of the functions h(A) (k = 1,2,3).

For the kernel R of the type (4.3.38), one has

8X~i X) = -1rt X(Ak)eF(),k) Sk(Ak,A)e- F(),) 6(A - Ak)6(X - Xk).k=l

Then, from the Cauchy formula (4.3.33), for A =1= Ai (i = 1, . . . , n) , one obtains

(4.3.39)

(4.3.40)

Now let us consider the Cauchy formula (4.3.33) with :~ given by (4.3.39) as A -+

Ai (i = 1, . .. ,n) . This yields

X(Ai) = 1 + 2~i JJd~: ~ ~~' ~ tX(Aj)eF(),j)Sj(A j, A')e-F(>")6(A' - Aj)6(X/ - Xj)'C 3=1

(4.3.41)The term in (4.3.41) with i = j is equal to

_ Resx(Ai)eF(>. ;) Si(Ai, A)e- F(>.) I(A - Ai)2 >'=),i

~ X(A,) ( a8,~~, At"",-8,(.1" A,)F' (A,))

8F(A)I . .where F'(A.) =~ . It follows that (4.3.41) gives rise to the system),=),i

(4.3.42)

X(Ai) +X(Ai)(SHAi, Ai)- s.o; Ai)F'(Ai))+L X(Ajl~~i..' Aj) = 1, (i = 1, . .. , n)j#i 3 •

(4.3.43)

h S' (\ \) (}Si(Ai ,A)! Th" f l' .w ere i Ai,Ai := (}A ),=),i ' 18 constitutes a system 0 n mear equations for

the n quantities X(Ai)' On solution of the system (4.3.43) for the X(Ai) (i = 1, .. . , n),the formula (4.3.40) then gives the function X(A, X). Knowledge of X()., X) allows us to

construct solutions of the corresponding nonlinear integrable equation.

(4.3.44)

Methods for Construction of (2+1}-Dimensional Integrable Equations 183

The dependence on x» (k = 1,2,3) in (4.3.43) arises due to the term F'(>'i) =

~ 8Ik(>')I' .~~ >.=>.~k' For this reason, all X(>'i) calculated from (4.3.43) are rational

functions of Xk (k = 1,2,3). The formula (4.3.40) showsthat the corresponding solutions

of the nonlinear integrable equation are likewise rational functions of Xk. This class of

rational solutions contains, in particular, the rational and lump solutions of the KP,

NVN-I, and other scalar equations. The system (4.3.43), (4.3.40) was first derived via

a nonlocal Riemann-Hilbert problem in [271] .

Nontrivial rational and lump-type solutions can be similarly constructed in the

matrix case. For this purpose, one chooses the matrix functions ik and gk in (4.3.30)

to be of the form(ik)a{J = AaI18(>' - >'(k)a{J)(A - A(k)a{J),

(gk)a{J = Ba{J8(>' - J.L(k)a{J)8(A - ii(k)a{J),

where Aa{J, Ba{J ,>'(k)al1' and J.L(k)a{J are complex constants. In particular, choosing the

ik and gk as diagonal matrices, we obtain

(4.3.45)

For cases when the nonlocal a-problem reduces to a local a-problem, nonlocal, or

local Riemann-Hilbert problem, the general a-dressing method likewise reduces to its

various specializations. It is noted also that the spatial transform method as proposed

in [302, 304, 507] is conceptually very close to the a-dressing method.

The a-dressing scheme described above can be generalized. The extension we have

in mind is connected with a generalization of the long derivatives D, proposed by

Zakharov [508]. He demonstrated that all the basic formulae (4.3.4)-(4.3.13) remain

valid if, instead of (4.3.3), one introduces generalized long derivatives of the form

8f (0) 8f (1) anf (n)Dil = 8Xi + f ~ (>.) + a>.~ (>.) + ... + a>.n~ (>.), (4.3.46)

where ~(k) (>') (k = 0, 1, . . . ,n) are matrices which commute in pairs and have a rational

dependence on >.. The inclusion of derivatives with respect to >. in D, allows us to

extend the class of auxiliary linear problems, and with it, the corresponding nonlinear

integrable equations. The case n = 1 with scalar functions ~(1) (>') corresponds to

spectral problems with a variable spectral parameter [43]. The case n = 1, but with

operators D, of the form

af t, afDil = aXi + f-y: + a>..:h, (i =-1,2 ,3)

184

where Ii and :li are constant N x N matrices, leads to the system [508]

Chapter 4

(i,j ,k = 1,2,3) (4.3.47)

where

Rij = :li(1- P) (:~ - f(.Ji + Ij)f) - (i +-+ j)

and P is an N x N matrix which satisfies the constraints p2 = P and P:liP = O. The

system (4.3.47) is nontrivial if N ~ 4. It represents an interesting generalization of

the system of equations (2.5.20) which corresponds to :li == O. The properties of the

system (4.3.47) and other integrable systems with n ~ 1 are markedly different from

the properties of systems with n = 0 as considered above.

A further generalization of the operators (4.3.46) leads to operators D, in the form

or

(4.3.48)

where :liCk)(XI,X2,X3,>') now depend also on of XI ,X2,X3. The operators Di given

by (4.3.48) give rise to integrable systems of a new type. The commutativity condition

[Di , Dk] = 0 is equivalent to a nonlinear integrable system for the coefficients:li(k)(x, >.).Thus, on use of (4.3.4), we obtain via (4.3.5)-(4.3.13) a new nonlinear integrable system.

This system may be regarded as an integrable system at the second level.

The a-dressing method admits, in principle, generalization to an arbitrary number

of independent variables Xl, . . . , xn . The spectral problems are constructed in the same

manner as for n = 3. These spectral problems are compatible by construction and,

as a result, lead to associated multidimensional nonlinear equations. For instance, for

arbitrary n, instead of three spectral problems (4.3.27) and three nonlinear equationsn(n - 1) n2(n - 1)

(4.3.28), we have 2 linear problems of the form (4.3.27) and 2 nonlinear

equations (4.3.28) for n matrix-valued functions Xi(Xl, . . . , z.,). For n > 3, the nonlinear

system (4.3.28) is highly overdetermined . Nevertheless, it has nontrivial solutions which

are parameterized by the functions R(>",A'; >.,A). The analysis of multidimensional

(n> 3) integrable systems constructed in this manner is complicated. The problem of

whether or not they are truly multidimensional remains open.

4.4. The a-dressing method with variable normalization

In the previous section, the a-dressing method was described for both fixed and

canonical normalization of solutions of the a-problem (4.3.1). However, as has been

Methods for Construction of (2+1}-Dimensional Integrable Equations 185

mentioned, other normalizations such as to rational functions are also admissible. The

different normalizations are not gauge equivalent . Here it is shown, following [298, 299],

that use of different normalizations allows the construc tion of explicit integrable systems

and their interrelations.

We introduce the nonlocal a-problem with general normalization 7](>'). This is

defined by the a-equation [298, 2991

ax a7]~=x*R+-=a>. a>.

with the boundary condition

X(>') - 7](>') -+ 0 as >. -+ 00,

(4.4.1)

(4.4.2)

where 7](>') is, in general, an arbitrary rational function of >.. For 7] == 1, equation (4.4.1)

coincides with the a-equation (4.3.1) in the canonical normalization.

Equation (4.4.1) represents a singular inhomogeneous a-equation. It can be rewrit­

ten in the form of an inhomogeneous a-equation for the regular function <p := X - 7]as

o <p-= = <p *R + 7] *R,a>.

(4.4.3)

where <p -+ 0 as >. -+ 00. This equation is uniquely solvable, at least for small R. In

what follows, we shall assume the unique solvability of the problem (4.4.1) - (4.4.2) for

arbitrary R. In analogy to the canonical normalization case, we have X == 0 by virtue

of the conditions aa~ = X * R and X ------ o.>. >'-00As in the previous section, it is assumed that the dependence of the function R on

the additional variables Xl ,X2 ,X2 is defined by equation (4.4 .3) . The gener al solution

of this equation is of the form

R(>.' >" . \ >.. x) = er::=1t, (>.')Xi Ro(>" >" . >. >.)e- r::=1li(>')xi, , A, , , , , , (4.4.4)

where Ro is an arbitrary matrix-valued function. In what follows, we consider the case

of rational I i(>') and choose Ro(>" , >" ;x, >') so that R(A', >" ;x,>.;x) has no essential

singularities at points where I i (>') has poles. For such Ro, the function R(>",>"; >., >.; x)

decreases at a .pole >'1 of I i (>') faster that any power (>' - >'1)n(>.' - >'dn for all >..Equation (4.4.1) also implies that ~r tends to zero faster th an any power (>' - >'dn at

the point >'1.

For the fixed canonical normalization X(oo) = 1, in order for the function MX to

be a solution of the a-equation (4.3.1) with the same normalization, it is necessary that

the condition (4.3.8) hold. However, in our case, since the normalization is not fixed,

this is not necessary.

186 Chapter 4

We now differentiate equation (4.4.1) with respect to Xi whence , in view of (4.3.2) ,

(4.4.5)

The last two terms in (4.4.5) always can be represented in the form :X JLi , where JLi

are appropriate functions. Hence, the functions DiX are also solutions of the a-problem

(a)-1 a

(4.4.1) but with different normalizations JLi = DiX - aX D, a~ '

It is now shown that, for rational functions Ii().) and 1]().) , the functions JLi().) are

likewise rational. To prove this, we use the formulae

and

a -n (_1)n-1 an-1

aX (). - ).0) = 7r (n _ 1)! a).n-1 o(). - ).0) , (4.4.6)

an n kaktp().O) an-ktp().) a).n o(). - ).0) = Len a).k a).n-k o(). - ).0) ,

k=O

whence it follows that the expression tp~i is the sum of a o-function and its deriva­

tives at the poles of I i ( )' ) . Moreover, t he formula (4.4.6) implies t hat t he expressions

(~) -1 tpa~~).) are rational functions. Accordingly, it may be shown that

(4.4.7)

so that the JLi are rational functions. Accordingly, the functions DiX are solutions of

the a-equation (4.4 .1) with rational normalization. The relation

gives

(4.4.8)

It follows from the latter relation and the fact that ~~ decreases faster t han any power

(). - ).t}n at poles of Ii().) that the functions DiX- JLi are regular. Therefore, t he JLi are

normalizations of the functions DiX. Thus, one can generate solutions of the a-problem

(4.4.1) via the operators D i . It is important to note that t he number of solutions which

can be constructed in this way increases with the order of the operators D; faster than

the number of the normalization poles. The differential equations for the solutions X of

the a-problem (4.4.1) are constructed in [298, 299).

Methods for Construction of (2+1)-Dimensional Integrable Equations 187

Let us now consider the general situation wherein the functions Ii(A) have an

arbitrary number of simple noncoincident poles so that

(i=I,2,3). (4.4.9)

For convenience, we introduce the notation I ~ (~) to indicate summation over 0

from 1 to n.

Let XI(X , A) be the solutions of the a-equation (4.4.1) normalized to (A - AI )-1 .

These functions admit the following expansions

(4.4.10)

The functions DiXy are also solutions of the a-equation (4.4.1). By virtue of (4.4.7),

the normalizations of DiXy contain first order poles at the points AI and Ay where y

is fixed and I is arbitrary. On the other hand, the normalizations of the functions XI

and Xy contain simple poles at the points AI and Ay , respectively. Accordingly, one can

construct a linear combination of the functions DiXy , XI, and Xy which has vanishing

normalization. This combination is

(4.4.11)

where Xyl := Xy(X ,Al) == x~~. In view of the unique solvability of the a-problem

(4.4.1), we have [298, 299]

(4.4.12)

along with the equations obtained by cyclic interchange.

Hence, in the general case (4.4.9), we obtain a system of linear spectral problems

(4.4.12) which are also linear in the Di. The ability to construct such auxiliary problems

linear in the D, is clearly connected with the use of rational normalizations of the

functions Xy' It is recalled that, in the case of the fixed canonical normalization X((0) =1, the spectral problems (4.3.26) are quadratic on D, even in the simplest case n =1. With increase in n, the order of the spectral problems increases quickly and their

construction becomes technically a more complicated problem. The use of a variable

rational normalization allows us to construct rapidly the spectral problem even in the

general case (4.4.9).

188 Chapter 4

The linear equations (4.4.12) are, in a sense we now demonstrate, basic to the a­dressing scheme under consideration. Thus, let XI be the solution of (4.4.1) normalized

to (>' - >'1)-1 and consider the class of the solutions of the problem (4.4.1) of the form

(LX)(x, >') = 2: U(I)n l ,n~.n3(X)D7 1 D?Di:3XI(X,>') + c.p.n1Jn~ .n3=O

(4.4.13)

where c.p. indicates cyclic permutation of the indices I, y, k. On use of the definition

(4.4.11) for Liy(X), one can show from (4.4.13) that

N

(LX)(x, >') = 2: UIn (x)DixI(X, >')n=O

N

+ 2: U(iy)nl,n~,n3 (X)D~l D'r Di:3Liy(x) + c.p.nl,n2,n3=O

(4.4.14)

Further, considering the poles of orders N ,N + 1, and so on, in turn, one can verify

that ifN

2: uln(x)DiXI(X, >') + c.p. = 0,n=O

(4.4.15)

then the left-hand side of (4.4.15) vanishes identically.

This result together with (4.4.14) implies that if L = 0, then L is representable in

the formN

Lx == L U(iy)nl ,n~ .n3(x)Di' D'r Dk' Liy(X) + c.p .n1 ,"2 ,"3=0

(4.4 .16)

Hence, any linear equation L(X) = 0, where the operator L is of the form (4.4.13),

represents, in fact, a linear superposition of equations (LiX)y = °of the type (4.4.12)

[298,299]. Thus, the spectral problems (4.4.12) may be regarded as fundamental objects

in the a-dressing method with variable normalization. The nonlinear system associated

with these spectral problems likewise plays a significant role and can be constricted in

a very simple manner. Thus, on expansion of the functions Xy(x, >') of (4.4.12) in the

series about the points >'k , we obtain the system [298, 2991

(4.4.17)

along with the equations obtained by cyclic permutation of the indices i, y, k. The

system (4.4.17) is the integrable system of 6n2 equations for 6n2 unknown functions

Xlk(X) associated with the fundamental auxiliary linear system (4.4.12) .

The system of nonlinear equations (4.4.17) occupies a central position within the

class of nonlinear systems integrable by the a-problem (4.4.1) . The solut ions of the

nonlocal a-problem (4.4.1) appropriately normalized give particular solut ion of the

Methods for Construction of (2+1)-Dimensional Integrable Equations 189

fundamental system (4.4.17). These solutions depend on the functional parameter

Ro(A',X'jA,X). The natural boundary condition for X (z) is that X (x) --+ (Ay -Iy Iy Ix1""'00

AI)-1 . In the case of degenerate Ro(>.', X'j A,X), the solutions of the system (4.4.17) can

be constructed in explicit form.

The fundamental system (4.4.17) is close in form to the system (1.3.10) descriptive

of three resonantly interacting waves. This allows us to establish that the system (4.4.17)

is Lagrangian. The corresponding Lagrangian is [298, 299]

AkAy _ AkXy/AIX1yAy

+ ~A/XlkAkXkyAYXYI - ~AIXlyAYXYkAkXkY }] ,

(4.4.18)

where sgn (ijk) indicates transposition of the indices i,j, k and summation is assumed

over the indices 1, 2, 3 and over Q for the common indices I.

The linear system (4.4.12) gives not only the fundamental system (4.4.17) but also

an infinite set of conservation laws for (4.4.17). To derive the latter, it is sufficient to

expand (4.4.11) in the neighborhood of the point Ay . The first two conservation laws

so obtained are local, while all the higher conservation laws are nonlocal.

The system (4.4.17) is of fundamental character in that it corresponds to generic

functions Ii(A) with an arbitrary number of simple non-coincident poles. Any rational

function Ii(A) can be obtained from such generic Ii(A) by an appropriate limiting

process. The corresponding nonlinear integrable systems can be likewise obtained as

appropriate limits of the fundamental system (4.4.17).

Let us now illustrate the interrelation between the fundamental system (4.4.17)

and different concrete integrable systems. First, we consider the fundamental system

(4.4.17) for n = 1. This is a system of six equations which under additional reduction

leads to the system descriptive of three resonantly interacting waves. In this instance,

the fundamental system turns out to be equivalent to the system (4.3.28), and the

interrelation between the funct ions Xij (X) and Xi(X) is given by the formula [298, 299]

() A-1 -1 (8Xj Ai Ai)Xij X = i Xi 8Xi + Xj Aj _ Ai - Aj _ Ai Aj , (i,j=I,2 ,3 j i =j= j ). (4.4.19)

190 Chapter 4

The simplest example of degeneration in the generic system (4.4.17) corresponds

to the case when the sets of poles for the functions li(>'), Iz(>') , and l3(>') coincide so

that

i = 1,2,3. (4.4.20)

Then it is convenient to introduce new functions Qkp(X) = Xkp - >. Ok\ . In terms ofp - k

these variables the fundamental system (4.4.17) adopts the form [298, 299]

(4.4.21)

where the indices p and q are fixed and there is summation over the remaining indices.

Note that the singularities in (4.4.21) cancel.

The nonlinear integrable system (4.4.21) for the n2 functions Qqr(X) is similar to

the resonantly interacting waves system. The Lagrangian of the system (4.4.21) is given

by (4.4.18) with the substitution X -- Qij'ly

The degeneration ofthe generic li(>') to functions li(>') with multiple poles is much

less trivial. The KP equation is a case where such a situation occurs [299]. If we make

the change>. -- >.-1 in (4.3.14), then within the framework of the a-dressing method,

the KP equation is associated with the operators

(4.4.22)

with the function X(x, >') normalized as X(x , >') -- >.-1 at >. -- O. These operators D,

contain multiple poles at >. = 0 and can be obtained out of a degenerate situation with

simple poles. Indeed, if one considers the operators

1D2 = 8y + 2e [(>' - e)-l - (>' + e)-I] ,

1D3 = D, + -2 [(>' + e)-l + (>' - e)-l - 2>.-1] ,

2e

then in the limit e -- 0, one retrieves the operators (4.4.22).

(4.4.23)

Methods for Construction of (2+1)-Dimensional Integrable Equations 191

The disadvantage of the operators (4.4.23) is that the corresponding h(A) contain

coincident poles. To remove these, it is convenient to perform the change of independent

variables (x, y, t) ---+ (x,~, "1) defined by

D~ = D 1 = 8x +A-I,

(4.4.24)D~ = ~(e2D3 + eD2 + Dt} := 8e+ (A _ e)-I,

D~ = ~(e2D3 - eD2 + Dt} := 8T/ + (A + e)-I .

The functions li(A) which correspond to the operators (4.4.24) have only simple poles

at the point 0, e,-e. Let us denote these poles by the indices 1, 2, 3.

The case (4.4.24) is a very special instance of the generic situation (4.4.9), and the

Lagrangian (4.4.18) takes the form

.c = X12 8T/X21 +X23 8xX32 +X318eX131 1 2

+ -2 X12X21 + -2 X13X31 + -X23X32e e e

+ X13X32X21 - X12X23X31 · (4.4.25)

Now, let us pass to the limit e ---+ 0. We will demonstrate, following [298, 2991 , that

the Lagrangian (4.4.25) reduces as e ---+ 0 to the Lagrangian associated with the KP

equation.

Firstly, wereturn to the fundamental equations (4.4.17) for the case (4.4.24). These

give

(4.4.27)

(4.4.26)

18eX13 + 2eX13 - X12X23 = 0,

18T/X12 - 2eX12 + X13X32 = 0,

whence, the functions X23 and X32 may be eliminated from the Lagrangian (4.4.25). As

a result, the Lagrangian (4.4.25) adopts the form

..c = xli (8eX13 + ;eX13)8x {X131(8

T/X12 - ;eX12)}

2 1+ 3(XI2X13)- 8eXI38T/XI2,

up to a total derivative. Further, if we introduce the functions 'Pij := Xij - Aj ~ Ai'

then the Lagrangian (1.4.27) becomes

2.c = -(8T/ln(1 + e'P12»8eIn(1 - e'P13)e

+'P12 + 'P13 + e8e'P13 8x {-~('P12 + 'P13) + e8T/'P12} . ( )

( )4.4.28

21+e12 -1 + e'P13

192

Now, we pass to the limit e -+ O. On use of the expansions

(1) (2) 2 (3) 3 O( 3)1t'12 = It'll + 1t'11 f + 1t'11 e + It'll f + e),

(I) (2) 2 (3) 3 O( 3)1t'13 = It'u - It'll f + It'u f - It'll f + e},

Chapter 4

with v := Xu (x, y, t) and subsequent elimination of It'g> by means of the relation

derived from (4.4.17), we obtain expansion of the Lagrangian (4.4.28) in the following

series in e:

(4.4.29)

The first nontrivial term in (4.4.29) is, in fact, the Lagrangian for the KP equation.

Indeed, it is readily verified that the corresponding Euler-Lagrange equation yields

and an appropriate rescaling of the independent variables leads to the KP equation

in u := Vx ' Thus , it has been shown that the Lagrangian for the KP equation may

be derived as a special limit of the fundamental Lagrangian (4.4.18). Other nonlinear

integrable equations can be reproduced in a similar manner as limits of the fundamental

system (4.4.17) (see [298, 299)). Overall, it has been seen that the a-dressing method

with variable normalization is basic to the study of the general structure of three­

dimensional integrable systems.

4.5. Operator representation of multidimensional integrable equations

To conclude our discussion of (2 + I)-dimensional integrable systems , we now

consider some properties of their operator representations. For the most part , the (2+1)­

dimensional integrable systems presented previously are equivalent to the compatibility

conditions for auxiliary systems of two equations

(4.5.1)

where £1 and £2 are, in general, matrix partial differential operators which involve

ax, ay , and at. The operator form of these compatibility conditions may be represented

by the commutativity condition

(4.5.2)

Methods for Construction of (2+1)-Dimensional Integrable Equations 193

or by the triad representation

(4.5.3)

or, finally, by the quartet representation

(4.5.4)

where "Y1 and "Y2 are appropriate differential operators. The progress from representa­

tions of the type (4.5.2) to representations (4.5.3) and (4.5.4) was driven by the desire

to construct nonlinear integrable equations associated with more and more complicated

operators £1 and £2 .

For each nonlinear integrable equation introduced, we have given an operator

representation with concrete operators £1 and £2 . It is clear , however, that if a given

nonlinear equation is equivalent to the compatibility conditions for a linear system of

the form (4.5.1), then there are many alternative linear systems

(4.5.5)

the compatibility conditions for which generate the nonlinear equations. Thus, it is

sufficient to rewrite the system (4.5.1) in any equivalent form, and we arrive at a system

(4.5.5) which gives the same nonlinear integrable equation.

The KP equation provides perhaps the simplest example of such non-uniqueness.

Thus, recall that the KP equation is equivalent to the compatibility condition for the

system (2.1.2) , namely,

£11/J = (aay + a-; + u(x, y,t»1/J = 0,

and that the operator form of this compatibility condition is

(4.5.6a)

(4.5.6b)

(4.5.7)

Let us transform the system (4.5.6) into an equivalent one. From equat ion (4.5.6a), it

is seen that

whence

(4.5.8)

194 Chapter 4

Bearing in mind that 1/J is a common eigenfunction of the operators LI and L2, substi­

tution of the expression for 48~1/J as given by (4.5.8) into equation (4.5.6b) leads to the

system

(4.5.9)- -1L21/J = (8t - 4(18x8y+ 2u8x - U x - 3(18x Uy)1/J = 0

which is evidently equivalent to the original system (4.5.6). The compatibility condition

for the system (4.5.9) is again the KP equation (1.3.2), but now we have the triad

operator representation

(4.5.10)

instead ofthe commutativity condition (4.5.7). Note that the second equation (4.5.9)

is nothing but equation (4.3.29) derived earlier via the a-dressing method.

A further example of this nonuniqueness is provided by the OS equation (1.3.5). In

the standard formulation, the OS equation is equivalent to the commutativity condition

for the operators (1.3.6). An equivalent pair of operators £1 and £2 can be constructed

as in the KP case. Indeed, the equation

(4.5.11)

(4.5.12)

gives

1 (1 0) 1 (0 q)~1/J=-- ~1/J-- ~a a -1 a -r a

On use of the latter expression for 8x1/J, the equation L21/J = 0 may be rewritten in theform

L21/J := (8t + .;; (~ ~b) o.o; + QI8y + Q2) 1/J = 0,

where Ql and Q2 are certain 2 x 2 matrices. The compatibility condition for the system

of equations (4.5.11) - (4.5.12) is again equivalent to the OS equation, but now the

operator form of the compatibility condition is as considered in [291, 377], namely,

(4.5.13)

Thus, the operators Ll and L2 for a given nonlinear integrable equation are not

uniquely defined. In view of this, it is natural to restrict the choice of the operators L,

and L2 by the requirement that the form of the general operator representation (4.5.4)

or, equivalently, of the equation

(4.5.14)

Methods for Construction of (2+1)-Dimensional Integrable Equations 195

with N, = -'Yl - L2 and N2 = L, - 'Y2 be conserved. It can be readily seen thatequation (4.5.14) is invariant under the transformation [272]

L i - L~ = L Qil"Lk,k=I,2

where the operators Qik and Qki obey the constraint

(4.5.15a)

(4.5.15b)

L QikQkl = 6il,k=I,2

(i,i = 1,2) . (4.5.16)

Hence, if a given nonlinear equat ion is representable in the form (4.5.14) with

certain operators L; and N«, then the infinite family of the operators L~ and n;connected with L i , N, by the transformations {4.5.15} likewisegenerate this equation. It

is natural also to require that, for differential operators L i , N, the transformed operators

LL Nf, be likewise differential operators. This condition is fulfilled if Qik and Qki are

differential operators. In what follows, we restrict ourselves to this situation.

For equation (4.5.4), the transformat ions {4.5.15} have the form

t; - L~ = L QikLk ,k=I ,2

'Yl - 'Y~ = L 'YQk! - LIQ21 - Q21Ll+k=I ,2

{4.5.17}

'Y2 - 'Y~ = L 'YkQk2 - L 1Q22+ QU L1+k=1 ,2

+ L2Q12 + Q12L2.

The general transformations {4.5.17} preserve the general equation {4.5.4}. There are

some important special cases. In particular, for Qii = Qii = 1 {i = 1, 2},Q12 = Q12 = 0,

and Q21 = -Q21 == Q, the transformations {4.5.17} reduce to

LI-L~=Ll, L2-L~=L2+QL1,

{4.5.18}

. ,'Y2 - 'Y2 = 'Y2,

where Q is an arbitrary differential operator. The transformation leaves L; and 'Y2

invariant. In the case 'Y2 = 0, one has

LI-L~=Ll , L2-L~=L2+QL1 ,{4.5.19}

196 Chapter 4

The transformations (4.5.19), as may be readily seen, preserve the form of the triad

representation (4.5.3). Such transformations (4.5.19) were first considered in [252] .

The flexibility in the form of the operators £1 and £2 as described by the transfor­

mations (4.5.15) - (4.5.19) can be exploited to choosethe operators £1 and £2 in the most

convenient manner. In particular, given operators £1 and £2 with a certain operator

representation, it is of interest to study possible equivalence to a simpler operator

representation with a smaller number of nonzero "(i . For instance, it is of importance

to ascertain for which £1 and £2 the quartet representation (4.5.4) is equivalent to a

triad representation (4.5.3) or, in turn, which triad representation is equivalent to a

commutativity representation (4.5.2). In this connection, one must determine for which

operators £1 and £2 it is possible to choose the operators Qik and Oki so that (4.5.17)

gives rise to "{~ = 0 or, correspondingly, so that in the transformations (4.5.19), one

gets "{~ = O. Such a procedure requires the solution of operator equations . It is simpler,

instead, to use the following rather obvious considerations. Let the operators £ 1 and

£2 be of the form

(4.5.20)

Since the commutator [£1, £2] is a differential operator of zero order in ay and at, it is

evident that "{I = "{2 = 0, that is [£1, £2] = O. Further, let

(4.5.21)

where L and v are now partial differential operators . In this case, the commutator

[£1, £2] is a polynomial in at of zero order so that "(2 = O. Hence, for operators £1 and

£2 of the form (4.5.21) we obtain a triad representation (4.5.3).

These arguments allow us to establish that, for operators £1 and £2 of the form

(4.5.22)

the triad representation (4.5.3) is equivalent to the commutativity representation (4.5.2)

with operators

(4.5.23a)

and

(4.5.23b)

Let us consider the transformation (4.5.19). In order for the operators £1 and £~

as given by (4.5.19) to commute, it is sufficient that the operator £~ not contain the

derivative ay • Since

(4.5.24)

Methods for Construction of (2+1)-Dimensional Integrable Equations 197

and v(8x ,8y) is a "polynomial" in 8y, there always exists a polynomial Q(8x,8y) in

8y such that the operator L2 is a polynomial in 8y of zero order. This Q(8x,8y) is

readily calculated for given v(8x,8y), and the order of Q(8x,8y) in 8y is lower by one

than that of the operator v(8x , 8y). Further, since the operator L2constructed in such

a way does not contain 8y at all, the replacement in (4.5.24) of 8y by -u(8x ) leads

to the operator (4.5.23b). Thus, we obtain the commutativity operator representation

[£11£2] = 0 with operators £1,£2 given by (4.5.23). In view of (4.5.19), this also means

that the operator 1'1 in the original triad representation is of the form 1'1 = [Q, Ld. It

is noted that one can retrieve the operator (4.5.23b) by simply eliminating 8y in the

operator L2 of (4.5.22) with the aid of the relation L1'l/J = 8y'l/J+u(8x)'l/J . This procedure

is equivalent to the special transformation (4.5.19).

An example of the above equivalence is provided by t he triad operator representa­

tion (4.5.9) - (4.5.10) for the KP equat ion and the usual commutativity representation.

In that case, Q = 48x . Similarly, the triad operator representation (4.5.12) - (4.5.13)

for the DS equation is converted into the usual commutativity representation by an

appropriate transformation of the type (4.5.19).

It follows from the above considerations that each of the nonlinear equat ions repre­

sentable in the form (4.5.2) with operators of the type (4.5.20) possesses an infinite

family of triad representations (4.5.3). Conversely, any operator representation for

such a nonlinear equation is equivalent to some commutativity operator representation

(4.5.2). It is natural to refer to such a commutativity representation as the irreducible

operator representation.

The same situation does not apply to the NVN equation (1.3.23) or equations

(1.3.24), (3.7.19), and (3.7.21). Thus, it may be verified that , for these equations , the

structure of the operator L1 = 8e8'1 + v8'1 + u precludes the existence of a differential

operator Q for which the expression 1'1 + [L1 , Q] vanishes. Accordingly, for the NVN

equation and equations (1.3.24), (3.7.19), and (3.7.21) the scalar triad representation

is the irreducible operator representation. Nevertheless, the possibility of constructing

a commutativity operator representation for these integrable equations exists. This is

connected with transition to 2 x 2 matrix operators [266] . Thus , as mentioned in Section

3.7, the linear problem

(4.5.25)

can be represented in the 2 x 2 matrix form

(4.5.26)

198

or, equivalently, in the form

Chapter 4

(4.5.27)

where

(4.5.28)

From (4.5.28), one can easily construct a 2 x 2 matrix form of the second auxiliary linear

problem L21/J = 0 for these equations, namely,

(4.5.29)

For instance, for equation (3.7.21), one has

(4.5.30)

and, for the NVN-I equation,

(4.5.31)

where the scalar operator L2 in (4.5.31) is given by (1.3.23) with k l = k2 = 1,a =

i,8T/ = a,8~ = 8.

The operator form of the compatibility condition for the system (4.5.27), (4.5.29)

is the triad representation

(4.5.32)

where, for equation (3.7.21),

M (0 0)/'1 = 2u~ v~

and, for the NVN-I equation,

Hence, the 2 x 2 matrix form of the auxiliary problems for the integrable equations under

consideration leads to a corresponding 2 x 2 matrix triad representation. However, the

2 x 2 matrix operator LfA given by (4.5.27) contains the pure derivative 8y • This allows

us to eliminate 8y from the operator L¥ by an appropriate matrix transformation

Methods for Construction of (2+1)-Dimensional Integrable Equations 199

(4.5.19). A 2 x 2 matrix operator Lr of the form (4.5.23b) is readily obtained. Thence,

we obtain a commutativity representation

(4.5.33)

equivalent to the matrix t riad representation (4.5.32).

Thus, the NVN equation and equations (1.3.24), (3.7.19), (3.7.21) also possess 2x 2

matrix commutativity representations (4.5.33) (see [266]) . The corresponding operator

Lr is given by (4.5.27), and the operator Lr, for instance, for equation (3.7.21), is

and, for the NVN-I equation,

The 2x 2 matrix commutat ivity representations (4.5.33) for other nonlinear systems

associated with the operator L 1 = 8~8Tj + v8Tj + u may be likewise constructed. These

commutativity representations are reducible, while the irreducible operator forms of

such integrable equations are given by scalar triad representations.

It is clear from the above considerations that matrix commutativity representations

exist for any system for which the auxiliary equation L, (8x , 8y )'l/J = 0 may be represented

in the matrix form Lr~ = (8y + uM (8x»~ = 0, where uM is some matrix differential

operator. It is noted that the more complicated problem of the equivalence of the

quartet representation to a triad representation can be analyzed in a similar manner.

The operator equations (4.5.2) - (4.5.4) discussed above are all representations

of nonlinear equations which inherit integrability from associated (1 + I)-dimensional

systems. Let us now turn to an intrinsically three-dimensional system. The nonlinear

system (4.3.28) (or (4.3.29» is representative of a class of such three-dimensional in­

tegrable equations. It may be shown that this class is equivalent to the system of the

operator equations [272]

(i, k,n,= 1,2,3; i =F k, k =F n, i =F n)(4.5.36)

200 Chapter 4

where the operator Lik is given by (4.3.27) with the functions U~k and U7k defined by

the formulae (4.3.25) and

(4.5.37)

where the indices i, k, n take the values 1,2,3, and here, there is no summation over

repeated indices. If we set L 12 ~ L 3 , L 13 ~ L 2 , and L 23 ~ L 1, then we arrive at the

operator system

(4.5.38)(i,k=I,2,3)3

[Li, Lkl =L "tikiLi,i=l

where the "structure constants" "tiki may be simply expressed in terms of Clink, {3ink and

Oink. For instance,

The fundamental system (4.4.17) represents another generic three-dimensional in­

tegrable system. In that case, we also have the three operators L, defined by (4.4.11),

but the operator representation (4.5.38) is commutative (all "tiki = 0).

The operator system (4.5.38) appears to be the most general operator representa­

tion of intrinsically three-dimensional integrable equations. The latter are equivalent to

the compatibility conditions for a system of three linear equations

L(l/) = 0, i = 1,2 ,3. (4.5.39)

The operator form of these compatibility conditions is (4.5.38). In general, the "struc­

ture constants" "tiki are differential operators. The main feature of the integration

procedure for these three-dimensional integrable equations is that one must solve si­

multaneously all three linear problems (4.5.39) in contrast to the procedure for the

"hereditary" (1 + I)-dimensional equations. In special cases, the general operator

representation (4.5.38) reduces to one of the operator representations (4.5.2)-(4.5.4). An

interesting problem is that of the equivalence of the representation (4.5.38) to a simpler

one. Another resides in the construction of irreducible representations of the type

(4.5.38) beginning with a commutativity operator representation for the fundamental

integrable system (4.4.17).

Methods for Construction of (2+1}-Dimensional Integrable Equations 201

The operator representation (4.5.38) can be regarded as a logical development of the

original notion of a commutativity operator representation. It reveals deep connections

between nonlinear integrable equations and closed algebras of differential operators.

The closed differential operator algebras defined by the system (4.5.38) can be

treated as a generalization of Lie algebras to the case when the structure constants

are differential operators. It is to be expected that such operators should, like Lie

algebras, play a fundamental role in theory of the integrable systems. Indeed, one can

seek to construct integrable equations directly from the operator system (4.5.38). For3 3

instance, if we set i; = L AiklOXkOXl +L UinOxn +Wi and require that the operatork,l=1 n=1

equations (4.5.38) hold, one obtains a system of nonlinear equations for the functions

Aikl(X), Uin(X), and Wi(X) , Operators of higher orders can be considered in a similar

manner. The disadvantage of this approach is the absence of guaranteed compatibility

of the resultant systems of nonlinear equations.

The operator representation (4.5.38) may be obviously generalized to the case of

an arbitrary number d of independent variables. Multidimensional (d ~ 4) integrable

equations which possess a commutative operator representation (4.5.38) (/ikl = 0) will

be considered in the next chapter. On the other hand , lower-dimensional (d = 2)

nonlinear equations with a representation (4.5.38) are also of interest. The general

operator representation of such two-dimensional integrable equations reduces to the

form

(4.5.40)

The system

(4.5.41)A'l/J - aep2 = 0

is a representative of this class of two-dimensional integrable equations [509]. Here

ep(x, y), 'l/J(x, y) are scalar functions , A = ao; - (Jo~ and a, (J are arbitrary constants.

System (4.5.41) may be represented in the form (4.5.40), where

1 1 3L1 = OxOy + 2epOy - 4:epy - 4:'l/Jxy,

2 2 a 3 2 3L2 = aox + (JOy - aepox + '2epx + 4: a ep - 2'l/Jxx,

/1 = -2aepx, /2 = -epy.

(4.5.42)

System (4.5.41) evidently can be rewritten as a single nonlocal equation by eliminating

the function 'l/J. Note that , for (J = 0, system (4.5.41) reduces to the simple system

epxy - 2ep'l/Jxy = 0,

202

or, equivalently, to the single equation

Chapter 4

Other examples of two-dimensional integrable equations can be obtained as the station­

ary limit (at -. 0) of (2 + I)-dimensional nonlinear equations which possess operator

representations (4.5.3), (4.5.4).

To conclude, we remark that noncommutative operator representations of two­

dimensional integrable equations have been discussed in [68, 325, 432] .

Chapter 5Multidimensional Integrable Systems

5.1. The self-dual Yang-Mills equation

In this last chapter, we consider certain interesting nonlinear integrable systems

which involvefour or more independent variables. The most important multidimensional

nonlinear equation integrable by the 1ST method is the self-dual (or anti-self dual)

equation of Yang-Mills theory, namely,

(IL, v = 1,2,3,4) (5.1.1)

(5.1.3)

where Fp.1I is the stress-tensor of the Yang-Mills field and F;II := ~fP.llprFpT is the tensor

dual to Fp.II' Here, fp.llpr is the usual antisymmetric constant tensor.

Gauge fields are basic to the modern theory of elementary particles (see, for ex­

ample, [441]). The main objects of gauge field theory comprise the nonabelian vector

field A~ and the stress tensor F:II , where IL, v are the Lorentz indices and the index a

corresponds to a local gauge group G (see [441]). In what follows, we restrict ourselvesto the case G = 8U(2) . The notation

3

Ap.:= ~L ~eTaA~,a=l

(5.1.2)3

Fp.II := ~ L ~eTaF:1Ia=l

is employed here as in [441] . The eTa (a = 1,2,3) are the usual Pauli matrices. The

8U(2) Yang-Mills theory is defined by the Lagrangian,

.c = 8~2 tr(Fp.IIFp.II)

where(5.1.4)

203

204

and the covariant derivative Vp. is given by

Chapter 5

(5.1.5)

where 8p. == 88 . The field equations which follow from (5.1.3) are of the formxp.

(5.1.6)

(5.1.7)

The Lagrangian (5.1.3) and the field equations are invariant under the gauge transfor­

mationsAp.(x) ~ A~(x) = GAp.G-1 + G8p.G-1,

Fp.II(x) ~ F;II(x) = GFp.IIG-1,

where G(x) is an 8U(2)-valued function.

An important feature of the gauge fields is that they possess a topological charac­

terization in terms of the topological charge (Pontryagin index)

(5.1.8)

This represents the degree of the mapping 8 3 ~ 8 3 determined by the vector potential

Ap.(x) (see [279]) .

The self-dual (or anti-self-dual) equation (5.1.1) arises in the analysis of the infra­

red behavior of the Yang-Mills fields [510] . The solutions of the Euclidean form of

equations (5.1.1) (with gp.1I = OP.II) minimize the action 8 corresponding to the La­

grangian (5.1.3). For such solutions, by virtue of (5.1.3) and (5.1.8), the Euclidean

action 8 is proportional to the topological charge Q. These features make the self-dual

equations (5.1.1) very important from the physical point of view. This is despite the fact

that their solutions form but a small subset of solutions of the full Yang-Mills equation

[276J.

The applicability of the 1ST method to the self-dual Yang-Mills equations (5.1.1)

has been demonstrated by Belavin and Zakharov [273, 274J. They have shown that the

equation Fp.1I = -F;II is equivalent to the compatibility condition for the linear system

Ll1/J = [>,(V4 - iV3) - (V2 + iVdJ1/J = 0,

L21/J = [>,(V2 - iVd + V4 + iV3J1/J ~ 0

with the commutativity operator representation

(5.1.9)

(5.1.10)

Multidimensional Integrable Systems

For the self-dual Yang-Mills equation, the operators L1 and L2 are [273, 274]

L 1 = ->'(V4 + iV3) - (V2 + iVd,

205

(5.1.11)

Equations (5.1.1) and (5.1.9) can be rewritten in a more compact form if, as in

[273, 274], one introduces the complex variables

(5.1.12)

The anti-self-dual equation (5.1.1) then yields

8%lB2 - 8%2Bl + [B1,B2] = 0,

L (8%IBt + 8i lBi + [Bi , BtD = 0,i= 1,2

while the linear system (5.1.4) becomes

L1'IjJ = (>'8%1 - 8i2+ >.Bl + Bt)'IjJ = 0,

L2'IjJ = (>'8%2 + 8i 1+ >'B2 - Bt)'IjJ = 0.

(5.1.13a)

(5.1.13b)

(5.1.14)

The operators (5.1.9), (5.1.11), or (5.1.14) are particular cases of the operators

(1.3.32) which are typical of the second method of the multidimensionalization discussed

in Section 1.3. The main characteristic of these operators is that they only contain

derivatives of the first order in the independent variables. In addition, these operatorscontain an explicit dependence on the spectral parameter which cannot be eliminated

by a transformation of the type 'IjJ --4 'IjJ'IjJo, where 'ljJo is the solution of the corresponding

linear problems with vanishing potentials.

Another important feature of the problem (1.3.22) and, in particular, of the prob­

lems (5.1.9), (5.1.11), and (5.1.14), resides in the fact that, in this case, the inverse

problem equations are generated by a local Riemann-Hilbert problem [273, 274] . In

this respect, the self-dual Yang-Mills equation differs from the integrable equations

considered in Chapter 2. If one starts with the local Riemann-Hilbert problem on an

arbitrary contour r, then from the formulae (1.2.13) - (1 2.18), one can construct broad

classes of solutions for equation (5.1.1).

The analogs ofthe formulae (1.2.17), (1.2.18) which follow from (5.1.14) are of the

form

(5.1.15)

206 Chapter 5

The system of equations (5.1.14) possesses an important symmetry property. Thus,

if 1/I(x, A) is a solution ofthe system (5.1.14), then the function [1/1+ (x,-j) ] -1 is also

a solution. Hence, it is natural to impose on 1/1 the following constraint [273, 274J:

(5.1.16)

On application of the general local Riemann-Hilbert problem method (see Sec­

tion 1.2), we obtain the following simple procedure for the construction of solutions

of the anti-self-dual Yang-Mills equation (5.1.1) [273, 274J. First, we take any 2 x 2

matrix-valued function 1/I(x, A) which obeys the condition (5.1.16). Then, we demand

that the function 1/I(x, A) be such that the expressions in the right-hand side of (5.1.15)

are linear functions of the spectral parameter A. The functions Bi(x) then constructed

via the formulae (5.1.15) give solutions of the anti-self-dual equation (5.1.1). A wide

class of solutions can be constructed in this manner.

(5.1.17)

_ { A- AO 1+ AAo - +}1/I(x, A) - v u+ 1 + AAo fA + A_ Ao fA,

1/I-1(X,A) = {u- 1\--A~ofA- lA~A;ooJA+l-:where u = u(x), f = f(x), and v is a 2 x 2 unitary matrix. The identity 1/11/1-1 = 1implies that A2 = 0 and

The solutions which are bounded throughout four-dimensional Euclidean space

and are the analogs of lumps are of the greatest interest . These correspond to functions

1/I(x, A) which have poles in the A complex plane. Here, we construct the simplest

solution of this type in the manner of [273,274] (see also [511]) . Firstly, it is seen from

(5.1.16) that the function 1/I(x, A) must have at least two poles located at the points

A = Ao and A = - ;0' By virtue of (5.1.16) , such a function 1/1 can be represented in

the form

(5.1.18)

The unitary matrix v in (5.1.17) is associated with gauge transformations of the gauge

fields. It may be shown that the fields B, are nonsingular only in the case Ao = O.

Hence, we put v == 1 and AQ =O. Further, it is convenient to parameterize the nilpotent

matrix according to

(5.1.19)A= lal2: Ibl2 (~:~ , ~:b)'where a and b are appropriate functions. For matrices A of the form (5.1.19) one has

AA+ +A+A = 1 so that the condition (5.1.18) reduces to

(5.1.20)

Multidimensional Integrable Systems

Hence, we choose the matrix 1/J(x, >.) in the form

1 - +1/J(x, >.) = u + >'1A + AI A ,

1/J-l(x, >.) = u - >'1A - ~1A+,

207

(5.1.21)

where the matrix A is given by (5.1.19) and the functions u and I are connected by the

relation (5.1.20) . The problem now is to find functions a, b, u, I such that the right-hand

side of (5.1.15) is a linear function of >.. Ifwe substitute (5.1.21) into the right-hand side

of (5.1.15), then the coefficients of>.-2, >. -1, >.2, and >.3 must vanish identically. These

conditions lead to the requirements

A8z;A = 0,

Jl + 1112 _ Zl (a8i 2b - bai 2a) - z2(a8i l b - bail a) + C

I - lal 2 + Ibl2 '

(5.1.22)

(5.1.23)

where C(Zl,Z2) is an integration constant. It follows from equation (5.1.22) that the

functions a and b are analytic in the variables Zl and Z2 Thus, we have three analytic

functions a(Zl,Z2), b(Zl,Z2), and C(Zl ,Z2) which together with I arc restricted by

the solvability condition for the algebraic equation (5.1.23). In this connection , since

Jl; 1112~ 1, it follows that equation (5.1.23) is solvable only when the modulus of

the right-hand side is greater than unity. This condition is very restrictive. In fact, it

may be verified that it is fulfilled only if a and b are linear functions and c is a constant.

A simple such solution is

(5.1.24)

with corresponding fields B 1 and B2 given by [273, 274]

(5.1.25)

For the vector-potential A,.., we have

x2 8gA,..(x)= 2 2 g-

1(X)-8 ' (/-L=1,2,3,4)P +x x,..

(5.1.26)

208

where3

g(X) = (x4+i2:Xkak)(x2)~k=l

Chapter 5

and x 2 = xlJx,..

Thus, we have constructed the simplest solution of the anti-self-dual Yang-Mills

equation (5.1.1) which is both rational and bounded everywhere. This solution is

nothing but the celebrated instanton first derived in the form (5.1.26) in [275] . The

general one-instanton solution corresponds to the general solution of equation (5.1.23)

and depends on eight parameters, namely, the scale variable p, the four parameters

of the displacement xlJ - xlJ + xolJ' and three rotation angles of the global 8U(2)

symmetry group. In the limit Ixi - 00, the vector-potential AIJ(x) for the instanton

solution reduces to the pure gauge expression g-l aag . Substitution of (5.1.26) intoxlJ

(5.1.8) gives Q = 1. Hence, the instanton is a lump solution of the self-dual Yang-Mills

equation with nontrivial topology.

The dressing method also allows us, in principle, to calculate general multi­

instanton solutions. To do this, one must again use the formula (5.1.15). It follows

from the condition (5.1.16) that the function 'l/J(x, A) must have an even number of

poles at the points Ai and - {i (i = 1, .. . ,N). If one demands that the right -hand

side of the relations (5.1.15) be linear in Athen, in principle, the corresponding 'l/J(X, A)

and, thereby, the potentials B; and B2 can be found. However, it is required to obtain

the general solution of a system of the quadratic algebraic equations. This prevents theconstruction of the multi-instanton solutions in explicit form [274, 511J .

Other methods of construction of N-instanton solutions based, in the main, on

various Ansatze have been proposed in [512-516J . Methods of algebraic geometry have

been developed for the solution of the self-dual equation in [276, 5171. These procedures

have been reported in a number of reviews and monographs (see, for example, [416,

525, 526]). It has been shown therein that the general N-instanton solution depends on

8N - 3 parameters. It is noted that an investigation of the self-dual equations close in

nature to that discussed in Section 4.1 has been proposed in [527] .

We next consider some algebraic properties of the anti-self-dual equation. It is

noted, first of all, that equation (5.1.13) can be rewritten in an interesting equivalent

form [528]. Thus, it follows from equation (5.1.13a) that

(5.1.27)

where ¢(Zl, Z2, Zl, Z2) is a 2 x 2 matrix-valued function and here ¢%' == aa¢ . We next• Zi

introduce the 8L(2, C)-valued function 9 according to [5241

g(Z,Z) := ¢(z)¢+(z). (5.1.28)

Multidimensional Integrable Systems 209

Since the gauge transformation (5.1.7) is equivalent to the transformation c/J -- c/J' = c/JG,

it follows that the quantity g given by (5.1.28) is gauge invariant. In addition, detg = 1

and g > O. In terms of g, equation (5.1.13) now adopts the form [529]

(5.1.29)

Hence, each solution of equation (5.1.29) generates a corresponding solution of the anti­

self-dual Yang-Mills equation and vice versa.

The interrelation between equations (5.1.13) and (5.1.29) is, in fact , very deep and

natural within the framework of the 1ST method. First, let us find an auxiliary linear

system whose compatibility condition gives rise to equation (5.1.29) . To construct such

a system it proves sufficient to substitute the expression (5.1.27) into (5.1.14) and then

introduce the function X = ¢'l/J. After some calculation, we obtain [529, 530, 531]

(5.1.30)

It is readily checked that the condition [£1, £2] = 0 does indeed give equation (5.1.29).

From the procedure for the construction of the operators £1 and £2, we obviously

haveA - - -1L, = c/JLic/J , (i = 1,2). (5.1.31)

Thus, the linear systems (5.1.14) and (5.1.30) and , correspondingly, the nonlinear

integrable equations (5.1.13) and (5.1.29) are gauge equivalent. In view of the relation

X = ¢'l/J, the normalizations of the local Riemann-Hilbert problems which correspond to

the problem (5.1.14) and (5.1.30) differ by the function ¢(z,z).

The variant of the self-dual Yang-Mills equation emloyed depends on the problem in

hand . Thus, in particular, the manifest gauge invariance of equation (5.1.29) makes that

form of the self-dual equation convenient for the construct ion of infinite sets of integrals

of motion, symmetries, and Backlund transformations [293, 525, 532, 552] . Note also

that the (2+ I)-dimensional MZM-I equation (3.3.1) and the linear problem (3.2.3) are

stationary versions with 8X3 -- 0 of equation (5.1.29) and the problem (5.1.30).

In some circumstances, a concrete parameterization of the 2 x 2 matrix g is useful.

For instance, if one parameterizes g by Poincare coordinates on the hyperboloid with

unit mass. That is, if one sets

(5.1.32)

210 Chapter 5

where C{), p are real and complex functions, respectively, then equation (5.1.29) is con­

verted into the system

(5.1.33)

(5.1.34)

This form of the self-dual Yang-Mills equation was first proposed in [528] . The advantage

of the form (5.1.33) is that all the constraints are now satisfied and C{), p are independent

fields. This elucidates the subsequent analysis of the system (5.1.33). For instance, one

can show that (5.1.33) is a Lagrangian system with Lagrangian .c given by [529]

L (C{)Zi C{)Zi + PZi lizi)

.c := ~ i=1,2 cp2

The properties of the self-dual equation in the form (5.1.33) have been studied in [529,

532].

The self-dual Yang-Mills equation has rather interesting geometrical properties

which follow from its representation as the compatibility condition of the linear systems

(5.1.9) or (5.1.14). The fact that the operators £1 and £2 contain only first-order deriva­

tives allows us, (ef. the MZM equation (Section 3.3», to introduce new independent

variables such that the potentials B 1 , B 2 are of pure gauge character. Indeed, in terms

of the variables1 Zl - >'Z2

VI = 2 >.

1 Z2 + >'ZlV2 = 2 >.

the system (5.1.14) adopts the form

_8.:...1/1.:...(V.:.;1l_V.:..:.2) + Fl1/1 = 0,8Vl

881/1 + F21/1 = 0,

V2

(5.1.35)

(5.1.36)

(5.1.37)

where Fl = >.Bl - Bt, F2 = >'B2 + Bt. The compatibility requirement for the system

(5.1.36) is the zero curvature condition [543]

8F2 8F1-8 - -8 + [Fl,F2] = °

VI V2

in the two-dimensional complex-space (VI, V2) . Equation (5.1.37) is, of course, equivalent

to the system (5.1.13).

Multidimensional Integrable Systems

The solution of equation (5.1.37) is clearly

211

(5.1.38)

where F( Vl, V2) is an arbitrary matrix-valued function of the variables Vl and V2 .

Accordingly, if we take any function F(vl, V2) such that the right-hand side of (5.1.38)

is linear in >., then we obtain solutions of the self-dual Yang-Mills equation. Hence,

the manifold of self-dual Yang-Mills fields is equivalent to a variety of pure gauge fields

in two-dimensional complex space. Such a description of the self-dual fields was first

proposed in rigorous form in [2761 .

A more precise geometrical formulation of the imbedding of self-dual Yang-Mills

fields into complex gauge fields in zero curvature has been given in [5171 . In summary,

this construction proceeds as follows. First, we introduce the four dimensional complex

projective space Cp3. For this purpose, we consider the four-dimensional complex

space with coordinates Yl,Y2, Y3, Y4· We then associate with these coordinates the new

variables Xl, X2, X3, X4 defined by

. Ydh + ihY4X4 + ZX3 = IY312 + IY412'

. Y2fh - ihY4X2 + ZXl = IY312 + IY412'

(5.1.39)

The variables xI" remain unchanged if one multiplies all the coordinates Yi by the same

factor. Hence, the formulae (5.1.39) define the projection of the three-dimensional

projective complex space Cp3 (six-dimensional real space) into the four-dimensional

real space R4•

The correspondence between the vector fields Ap.(x) in R4 and Ci(y) in CP3 is

given by

Correspondingly, the stress tensor

(5.1.40)

is expressed via the stress tensor Fp.v of the vector-potential Ap.(x) by the formula

(5.1.41)

212

From (5.1.39), t he quantity

D{x~,x,,) _ (ax~ ax" ax" ax~)D{Yi,Yk) = aYi aYk - aYi aYk

has the important property

Chapter 5

(5.1.42)D{x~ ,x,,) 1 D{xp , x.,.)D{Yi,Yk) = 2€~"pr D{Yi' Yk) .

Let us now consider the anti-self-dual field F~" = -F;". For such a field, from (5.1.14),

(5.1.43)

In view of (5.1.42), it is now seen that Fik = 0. Thus, the anti-self Yang-Mills fields in

R4 are in correspondence with the gauge fields in C p3 with zero stress tensor, namely,

the fields with the pure gauge potential O, = aa'l/; '1/;-1 , where the matrix-valued functionu.

'I/; is a homogeneous function of the coordinates Yi, ih (i = 1,2,3,4). Hence,

(5.1.44)

and it can be shown that t his system is equivalent to t he linear system (5.1.9). This

equivalence also assigns a geometrical meaning to t he spectral parameter A = Y3 (seeY4

[511, 517)) .

The rich geometrical structure of the self-dual Yang-Mills equation is detailed in

[517, 526]. Note that , whereas the special properties of the self-dual equation are

closely connected with its complex structure, reducibility to the zero curvature condition

(5.1.37) is a property common to the nonlinear integrable equations obtained via the

second multidimensionalization method (see also [553]) .

An important property of the Yang-Mills and {anti)-self-dual Yang-Mills equations

is their invariance under the conformal group [554]. The conformal group in four­

dimensional space also includes, besides the well-known Poincare t ra nsformat ions x~ --+

x~ = A~"x,,+a~, the dilation x~ --+ x~ = px~ (-oo < P < (0) and the special conformal

transformation x~ --+ x~ = 1+~(;X~:X~2x2 ' where C~ is an arbitrary 4-vector. It is

readily seen that t he linear systems (5.1.9) or (5.1.14) are not covariant under conformal

transformations. The auxiliary linear system for the anti-self-dual Yang-Mills equat ion

in the manifestly conformal covariant form has been set down in [555]. It is

Ll ;P = ({JLZ2+ AZI - b)D1 + {AZ2 - JLZl - a)D2) ;P = 0,

L2;P = ({JLZI - AZ2 + a)fh + {JLZ2 + AZl - b)D2) ;P = 0,(5.1.45)

Multidimensional Integrable Systems

where

213

and (a, b,A, JL) are arbitrary parameters. The system (5.1.45) can be obtained simply by

conformal transformation ofthe system (5.1.14). The set of parameters A = (a, b, A, JL)

is transformed under the conformal group in a nontrivial way. Thus , one can show that

A is a projective Cp3 twistor [555, 556].

The anti-self-dual equation is equivalent to the compat ibility condition for the

system (5.1.45). The operator form of this condition is [555, 556]

(5.1.46)

Hence, the manifestly conformal invariant operator representation of the anti-self-dual

Yang-Mills equation is a quartet representation of the type discussed in Section 4.5.

The representation (5.1.46) can be transformed, however, to the usual commutativity

representation [L 1, L2] = 0 by a conformal transformation. The conformal properties

of the system (5.1.45), as well as symmetries and Backlund transformations for the

anti-self-dual equation, have been discussed in detail in [556] .

The work that has been presented here has, in the main, concerned the anti-self­

dual Yang-Mills equation F,.." = -F;". However, analogous results hold for the self­

dual equation F,.." = F;". These results admit a straightforward generalization to

the case of an arbitrary classical, semi-simple Lie group G (see [416]) . Moreover, all

the constructions presented above can be carried over to the case of four-dimensional

space with arbitrary matrix g,.." and, in particular, to Minkowski space. However, an

essential difference arises in the construction of the rational solutions . Thus, rational

solutions which are everywhere bounded (instantons) exist only in Euclidean space.

Finally, there exists a supersymmetric generalization of the self-dual equations [557J .

The corresponding auxiliary linear system will be discussed in the next section.

5.2. The supersymmetric Yang-Mills equation

The proof of the integrability of the self-dual Yang-Mills equat ion represents a

remarkable achievement of the 1ST method. However, the full Yang-Mills equations are

of much greater interest. As already mentioned , gauge theories playa fundamental role

in modern elementary particle physics. For this reason, the discovery of a systematic

method of construction of physically reasonable solutions of the full Yang-Mills system

would be of the exceptional interest . However, all attempts at imbedding the full Yang­

Mills equation into known 1ST schemes have been, as yet , unsuccessful. Indeed, in the

rigorous Liouville sense the full Yang-Mills equations do not seem to be a completely

214 Chapter 5

(5.2.1)

integrable system. This fact has been demonstrated via computer methods in [558­562J. Nevertheless, this does not necessarily mean the the Yang-Mills equations are not

integrable in some wider sense.

The special nature of the algebraic-geometric structure of the full Yang-Millsequa­

tions has been described in [563, 564J. Witten [563J, in particular, has proved that the

Yang-Mills equations admit a simple imbedding in eight-dimensional complex space.

Let Yp, Yv (J.t, v = 1,2,3,4) be the coordinates of this eight-dimensional space as and

consider, as in [563J, the gauge field (Ap , Bv ) in as which obeys the system of the

first-order equations

1[V'y", V'y...] = - '2€pvpT [V'YP' V'Y-r],

[V'y", V'YvJ = 0,

where V'"" := 8"" + Ap(Y,Y), V'Yv := 8yv + Bv(Y,Y). Thus, the field Ap is a self-dual

field as a function of the variables Yp, while the field B; is anti-self-dual as a function of

the Yv ' It turns out that the Yang-Millsequations in Minkowski space are consequences

of equations (5.2.1) (see [563]). Thus, let us introduce the variables xp = ~(YP + yp)

and wp = ~(YP - yp). One has V'x" = V'y" + V'y" . Equations (5.2.1) and the Jacobiidentity imply that

As a result, on the diagonal YP = 'Up, one has

(v = 1,2,3,4). (5.2.2)

(5.2.3)

namely, the Yang-Mills equations (5.1.6).

Hence, the full Yang-Millsequations are a four-dimensional projection of the eight­

dimensional system of equations (5.2.1) which represent a particular superposition of

the self-dual and anti-self-dual equations [563J (see also [565]). The system of equations

(5.2.1) is equivalent, on the other hand, to the integrability conditions for the Yang-Mills

equations on certain four-dimensional surfaces in the eight-dimensional space (YP' 'Up) .The set of such surfaces is equivalent to ap3 x ap3 [563J. This fact allows us to give an

algebraic-geometry description of the manifold of solutions of the Yang-Mills equations

[563, 564]. The language of twistors has turned out to be very convenient for thispurpose [566, 567, 568J.

A similar construction also exists for the supersymmetric Yang-Mills equations

[563J. In contrast to the pure Yang-Mills equations, the corresponding auxiliary linear

problem has also been found (see [279, 519, 563]).

Multidimensional Integrable Systems 215

The supersymmetric Yang-Mills model contains not only the gauge vector field

A,..{x) but also spinor and scalar fields (see, for example, [570-574]). This model is

invariant under the so-called supertransformations which mix the boson and fermion

fields. It is convenient to formulate supersymmetric theories in a superspace [570-574].

This superspace is defined as the extension of the usual Minkowski space by the four

Grassmannian (anti-commuting) variables (JOt, ij/l (a, /3 = 1, 2):

[x'" ,(JOt] = 0, [z", ij/l] = 0,

ijit ij/l + ij/lijti = 0,

Thus, the coordinates in the superspace are za = (z", (JOt, ij/l). Here, a and f3 denote

spinor indices (a, /3 = 1,2) . One defines the superfields as functions of x", (JOt, ij/l on the

superspace. The supertransformations adopt the form

(5.2.4)

where ~Ot and ~/l are arbitrary spinors. The vector superfield is defined as the superfield

with the following transformation law

under the supertransformations (5.2.4). The superderivatives are of the form

D- a .(JOt aP· = - -=-:- - ~ p.aoP Ot ,

where aOt/l = (a")Ot/l a~,., aO = 1, and a i are Pauli matrices.

The N-extended supersymmetry is defined in a similar manner [570-574]. In this

case, the superspace has the coordinates (z", (}e;, ijiJt ) where J.L = 1,2,3,4; /3 = 1,2;

5, t = 1,2, .. . ,N and 0';, ij/lt anticommute. The 2N operators

tr: a .(}-/lsaOt = a(}Ot + ~ Ot/l'

sD- a ·(}Ota

/It = - aij/lt - ~ t Ot/l (5.2.5)

are analogs of the derivatives DOt and D/l.

The covariant derivatives are defined as follows:

\lOt/l = aQ/l + AQ/l'

\l~ = D~ + A~, V/lt = D/lt + Apt·(5.2.6)

216 Chapter 5

(5.2.7)

The super-vector-potential is A = (Aa,B' A~, A,Bt). The components of the superstress

tensor are given by

FOts,,Bt = [VOts, V,Bt]+,

F;a = [V,.., V~]_, F,..,,Bt = [V,.. , V.8t]-,

F~,.8t = [V~, V.8t]+ - 2i6;Va .8'

where [A, B]± := AB ± BA. The super-stress tensor defined in such a way contains

superfluous components which can be eliminated via the constraints [570-574]

F~~+F;~ = 0,(5.2.8)

F. i:>t+Fi:> ' t =0,os ,,.., ,..,8,Q

In the simplest case N = 1, the Yang-Mills equations are of the form

The super-Yang-Mills equations with different N are essentially distinct. It turns out

that only the cases N = 1,2,3,4 are possible . The cases N = 3,4 are of the special

interest since they are equivalent. For N = 3,4 the constraints (5,2,8) imply equations of

motion [572-574]. The N = 4 super-Yang-Mills model is associated with the Lagrangian

[569, 572-574]

where .,p is a set of four fermion fields and .,p is a set of six scalar fields.

The auxiliary linear system which gives rise to the constraints {5.2.8} for the case

N = 3 has been set down by Volovich in [269, 279] and adopts the form

Li.,p := {Vi + AV2}.,p = 0, s = 1,2

i = 1,2 (5.2.9)

It may be verified that the compatibility conditions for the system (5.2.9) are indeed

equivalent to the constraints (5.2.8). The system {5.2.9} which represents an auxiliary

linear system for the N = 3 super-Yang-Mills equation, by virtue of the equivalence of

theories for N = 3 and N = 4, also delivers the N = 4 super-Yang-Mills equations.

Multidimensional Integrable Systems 217

The operator representation of t he compatibility conditions for the system (5.2.9)

takes the form [279, 5691

[Li, L~]+ = 0, [L28 , L2i]+ = 0,

(5.2.1O)

The operators Li and L2i are odd, whence the appearance of the anticommutators

in (5.2.1O) . Although the operator representation (5.2.1O) is itself noncommutative, it

is equivalent to a commutative representation. To construct the latter, it suffices to

take 812tP as given by the third equation (5.2.9) and substitute it into the first two

equations. As a result, we obtain new operators ti ,t 2i, and L3 • It can then be verified

that equations (5.2.8) are equivalent to the commutative operator system [569]

In analogy to the case of the self-dual Yang-Mills equations, one can solve certain

of equations (5.2.8) by introducing auxiliary functions . The remaining equations (5.2.8)

then become a system of equations for t hese quantities similar in form to (5.1.29) (see(575)). This system represents the compatibility condit ion for five linear equations. The

corresponding linear operators are gauge equivalent to the operators L1,L2i , L3 given

by (5.2.9).

The linear system (5.2.9) and the corresponding local Riemann-Hilbert problem can

be used to reveal features of the super-Yang-Mills equations such as their symmetries

and Backlund transformations [279, 569, 575-579]. However, it should be noted that,

whereas the super-Yang-Mills equations reduce to the pure Yang-Mills equations (5.1.6)if all the additional spinor and scalar fields vanish identically, the system (5.2.9) becomes

trivial under such a reduction and does not deliver any information about the pure

Yang-Mills equations.

The super Yang-Mills equations are invariant under the superconformal group

SU{2, 2; N) [572-574]. The superconformal covariant auxiliary linear problem for the

super Yang-Mills equations was studied in [580-5811.

In the supersymmetric case, one can also introduce t he super-self-dual equations

[557]. These areFair = 0, N = 1,

(5.2.11)Fa 8 ,pi = 0, N > 1.

218 Chapter 5

The linear system whose compatibility conditions are equivalent to the constraints

(5.2.8) and equations (5.2.11) is of the form [279, 565]

(Vi + "V 2)1/7 = 0,(5.2.12)

(V1i + "V2i)1/7 = 0,

(V 12 +"V2~J1/7 = 0.

An analogous system exists for the super-anti-self-dual equations [569]. It is readily

seen that the system (5.2.12) is a supergeneralization of the linear system (5.1.19).

In conclusion, we remark that investigation of the super- Yang-Mills equations based

on the use of algebraic geometric constructions (twistors, supertwistors, etc) is more

developed at present that the local Riemann-Hilbert approach (see, for example, [526,

582-584].

5.3. Multidimensional integrable generalizations of the wave,

sine-Gordon, and self-dual equations.

The three resonantly interacting wave model considered in Section 3.1, is another

example of a nonlinear integrable system ostensibly with four independent variables.

The generalization of this system to the case of N2 - N resonantly interacting waves in

three-dimensional space is of the form (1.3.35) with d = 3. The general multidimensional

system of this type is given by (1.3.35) with arbitrary d [277, 41OJ.

The system (1.3.35) concluded our list of multidimensional nonlinear integrable

systems equivalent to the compatibility conditions for a system of two linear equations.

As we already mentioned , the integrability of such equations is inherited from associated

(1 + I)-dimensional equations. Multidimensional nonlinear integrable systems which

are not hereditary are of considerable interest. Such systems are equivalent to the

compatibility conditions for more than two linear problems

i = 1, . . . n (5.3.1)

where L; are partial differential operators. In the generic case, the number n of equations

(5.3.1) coincides with the number d of independent variables and the operator form of

the compatibility condition is

d

[Li, Lk] = L 'YiklLt,t=1

(i,k=I, . .. ,d) (5.3.2)

Multidimensional Integrable Systems 219

where Tiki are appropriate differential operators. The three-dimensional integrable

system representable in the form (5.3.2) has been considered in Section 4.5. A particular

case is given by the system (4.3.28) with d ~ 4.

Multidimensional (d ~ 4) integrable systems of another type which possess the

commutativity operator representation (5.3.2) with Tiki = 0 have been constructed in

[278,585-591]. One example is the system [585-587]

3.- (_1_ 8a ti ) __1_ 8ali 8alk _ 08Xk alj aXj alkalj 8Xk aXj - ,

8ajk aji 8alk _ 08Xi - ali 8Xi -

(i l' j)(5.3.3)

{i 1'j,i l' k,j l' k}

(i l' k, k l' i ,i l' j).

Here, (aij) is an n x n orthogonal matrix with entries functions of the n independent

variables Xl! . . . ,Xn . The parameter € takes values e = 0 or € = 1.

In the simplest case n = 2, the orthogonal matrix possesses the well-known para­

meterization

(U . U)cos- sm-2' 2

a= u u'-sin- cos­

2 ' 2

and the system (5.3.3) reduces to the single equation

(5.3.4)

Hence, in the simplest case n = 2, the system {5.3.3} for € = 1 yields the sine-Gordon

equation and for e = 0 the wave equation U X t X t - U X 2 X 2 = O. Accordingly, it is natural

to refer to the general system {5.3.3} with e = 0 as the generalized wave (GW) equation

and the system (5.3.3) with e = 1 as the generalized sine-Gordon {GSG} equation [586,

587].

The system (5.3.3) was derived in [585-587] via the classical theory of surfaces of

constant negative curvature [168]. Thus, the GSG equation represents, in fact, the

Gauss-Codazzi-Ricci equations which describe the imbedding of hyperbolic n-dimen­

sional manifolds Mn with constant curvature unity in {2n - I)-dimensional Euclidean

spaces R2n-1 [585-586]. In a similar manner, the GW equation {5.3.3} arises in the

description of manifolds Mn of constant sectional curvature k < 1 which are contained

in the unit spheres s2n-1 [587] .

220 Chapter 5

The system (5.3.3) has been analyzed within the framework of the 1ST method

[287, 588]. It is equivalent to the compatibility conditions for the system of n linear

2n x 2n matrix problems [287]

(i = 1, .. . , n) (5.3.5)

where

Here, iii and 'Yi are n x n matrices of the form

(5.3.6)

and

iii = (f -1) el aei + aei ,

where (ei)kl = OikOil, (i, k,f = 1, .. . ,n). The case 0 = J.l := ..\ (..\ E C) corresponds to1 1

the GW equation (e = 0), and the case 0 = 2("\ + ..\-I),J.l = 2("\ - ..\-1) corresponds

to the GSG equation. The operator form of the compatibility conditions for the system

(5.3.5) is

i,j = 1, .. . , n.

It proves convenient to consider the GW and GSG equations separately. We startwith the GW equation. In this case, the auxiliary linear system (5.3.5) can be converted

by the gauge transformation t/J(x,..\) -1j;(x,..\) - g-l(x)t/J(x, ..\), where

1 (a(x) , -a(x))g(x) =!2 1, 1 '

into the canonical form

(i=I, . . . ,n) (5.3.7)

with

(

eos; = ~

o ) 1 ( b,+ 'Yi ,, Qi(X) =--ei 2 -bi + 'Yi,

-bi + 'Yi)

bi + 'Yi(5.3.8)

and bi = _at{}8a .Xi

Each of the linear equations (5.3.7) itself represents a one-dimensional 2n x 2n

matrix spectral problem in which the other coordinates arise as parameters. The

integration problem for the GW equation requires the simultaneous solution of all these

n linear problems.

Multidimensional Integrable Systems 221

We present here the salient stages of this procedure, as presented in [287] . First,

we introduce the new function X defined by X(x, A) = 'I/J(x ,A)e- AX..1, where x,.J :=n

L e.a; This function X obeys the system of equations,=1

aX-a - A[3i,X] - Q,(x)X = 0,x,(i = 1, . . . , n) (5.3.9)

each of which is one-dimensional. Further, it follows from (5.3.9) that the equality

holds for any two solutions Xl and X2 of the system. Hence, there exists a 2n x 2n

matrix function W(A) such that

(5.3.10)

Moreover, W(A) is a diagonal matrix for purely imaginary A. Thus, we may associate

with the system (5.3.9) the local Riemann-Hilbert problem

ImA = 0, (5.3.11)

where the imaginary axis iR plays the role of the conjugation contour r . Recall that

X+ and X- are the boundary values of the function X(A) analyt ic everywhere outside

the conjugation contour. The normalization of t he function X is the canonical one

(X --+ 1).A-+OO

The inverse problem equations are given by the standard local Riemann-Hilbert

problem equations (1.2.14) - (1.2.15) and by the reconstruction formula

Q,(x) = - lim A[3i,X(x,A)]A-+OO

(5.3.12)

which follows directly from (5.3.9). The matrix R(A) embodies the inverse problem data

for the problem (5.3.9).

An important feature of the linear system (5.3.9) resides in the necessity to consider

initial data defined on straight lines in Rn but not on (n-l)-dimensional surfaces [287].

Let ii be the unit vector in R" and v the vector orthogonal to ii. Let us define the

restricted version Xof the function X(x, A) to the straight line v+ sii (s E R) by the

formula [287]

X(s ,A) := X(s, A, n, v) = X(v+ sii,A).

For functions Xof the form (5.3.13), the system (5.3.9) redu ces to

~: - A[.1n X] - Q(s,n)x = 0,

(5.3.13)

(5.3.14)

222

wheren

a; = n..J = L «a;i = l

n

Q(S, it) = Q(S, n, v) := L ~Qi(V+ sit).i=l

Chapter 5

(5.3.15)

Hence, for fixed it we retrieve the usual one-dimensional spectral problem. The solutions

of the corresponding forward and inverse problems in the general position case are known

[94, 301]. The general situation corresponds to all distinct 2n quantities ±~. We will

refer to the direction it as the generic one if the 2n numbers ±ni are distinct.

By varying it in (5.3.14), we obtain an infinite family of one-dimensional spectral

problems. In view of this, it is natural to specify the initial data and their behavior

along the generic directions it. The results of [301] can be used to establish the unique

solvability of the Riemann-Hilbert problem (5.3.11) for a class of potentials which

decrease rapidly along the generic directions [287] . In this case, one also has

lim X(i/ + sn, >') = 1.8-+ <Xl

Thus, for the linear system (5.3.9) (or (5.3.7)), the well-posed initial-value problem

requires specifying the initial data along lines in H" instead of on (n - I) -dimensional

manifolds.

Now, let us consider the GSG equation. In this case, the system (5.3.5) is not

convertible into canonical form. Nevertheless, it can be reduced to a form which leads

to a lo~al Riemann-~ilb(e~ pr~~e)m. To do this, we perform the gauge transformation

'l/J = g'l/J, where 9 = In . As a result, the system (5.3.5) becomes [287]v2 1, 1

AA (a ) ALi'l/J = aXi - :Ii(>') - Qi(X, >') 'l/J = 0, (i=I, ... ,n) (5.3.16)

1 1 A A Awhere :Ii(>') := '2(>':Ii + ~:Ii), :T1 := :T1, :Ii := -:Ii (i = 2, .. . , n), and :Ii is given by

the formula (5.3.8), while

1 ( bi + "ti,Qi(X, >') = -2

-i. + "ti ,

-s.+ "ti) 1 -1 A+ 2\ (g Big - :Ii).bi + "ti 1\

(5.3.17)

The solution of the system (5.3.16) with a = 1,"ti = 0 is exp(x..J(>')). On introduction

of the function X(x, >') = ;j;(x,>') exp( -x..J(>.)) , we obtain the problem

ax-a - [:Ii (>') , X]- QiX = 0,Xi

(i=I , . .. ,n). (5.3.18)

Multidimensional Integrable Systems 223

Again, we consider the family of lines v+ sri (s E R), where ii is a unit vector in

R:' and the versions Xof the functions X restricted to these lines are defined by

x(s, A) := X(v+ sri, A).

For the function X, we have

where

and

~; - [In(A), X]- OX = 0,

n

In(A) := :Lni.Ji(A)i=l

n

O(s, A) = :LniQi(v+ sri, A).i=l

(5.3.19)

(5.3.20)

Hence, we arrive again at a family of one-dimensional spectral problems parame­

terized by the unit vector ii. The system (5.3.18) and the class of problems (5.3.19)

differ from the canonical systems (5.3.9) and (5.3.14) only in the form of the dependence

on the spectral parameter A. Thus, as .in the GW case, the inverse problem equations

for the systems (5.3.18) are generated by a local Riemann-Hilbert problem [287]

(5.3.21)

However, the conjugation contour r is now more complicated in that it is the union of

the imaginary axis and the unit circle IAI = 1. The form of the contour r is defined,

as usual, by the condition that the expression X,.J(A) be purely imaginary for generic

n. The contour T consists of five parts. The Riemann-Hilbert problem involves the

construction of a function X(x, A) which is analytic outside r and whose boundary

values on each of the parts of r are connected by the relation (5.3.21). The solution of

this Riemann-Hilbert problem is given by the standard formulae (1.2.15) albeit with the

rather complicated contour r. The function R(A) embodies the inverse scattering data.

It has been established in [287] that this Riemann-Hilbert problem has a unique solution

within a class of potentials which decrease rapidly along the lines v+ sri (s E R).

As for the GW equation, the well-posed initial value problem for the GSG equation

involves specifying the initial data on a family of straight lines in R:' [287]. It is noted

that the formulation of the initial value problem for the GW and GSG equations is

essentially different from the standard initial value problems in (2 + I)-dimensions as

discussed in Chapters 1 and 2. It should also be emphasized that the GW and GSG

equations considered above are illustrative of multidimensional integrable systems for

which the problems (5.3.1) must be solved simultaneously. The multidimensional gen­

eralizations of the self-dual Yang-Mills equations considered in [589] belong to the same

224 Chapter 5

class of integrable systems . These equations may be constructed as the compatibility

conditions for the linear systems

Lat/J := Vt{A)\7p.t/J{x, A) = 0, a= 1, . . . ,n (5.3.22)

where \7p. = 8x,. + Ap.{x), Ap.{x), J.L = 1,. .. ,d is the vector potential, and Vt are

polynomials in A. The compatibility condition for the system (5.3.22) is [La, LbJ = 0,

that is

(5.3.23)

where Fp." = [\7p., \7"J. The analysis of equation (5.2.23) allows us to describe various

multidimensional integrable generalizations of the four-dimensional self-dual equation

Fp." = F;" [588J. These involve systems of algebraic equations for the components of

the stress tensor Fp.".

Another method for the construction of multidimensional generalizations of equa­

tion (5.1.29) has been developed in [591J . This involves the system of n linear equations

(i = 1, .. . , n) (5.3.24)

in the n-dimensional complex space (Z1, . .. ,zn), where the operators D, are given by

(i = 1, .. . ,n) (5.3.25)

and Si = (-1)', zn+t == Zl. The compatibility condition [Li, Lkl = 0 for the system

(5.3.24) yields

Equation (5.3.26a) gives

8Ai 8Aj-8 - -8 + [Ai ,AjJ = 0,

Zj Zi

S . 8Ai _ S; 8Aj - °} . - .

8z j +t 8Zi+1

(5.3.26a)

(5.3.26b)

(5.3.27)A ( ) _1 89i Z =9 -8'

Zi

where 9 is a GL{N, C)-valued function . On substitution of (5.3.27) into (5.3.26h), we

obtain [591J

(i, j = 1, . .. ,n). (5.3.28)

For n = 2, the system (5.3.28) leads to equation (5.1.29). The multidimensional system

(5.3.28) (n ;::: 3) can also be considered as a higher-dimensional integrable generalization

of the self-dual Yang-Mills equation. In [591J, the system (5.3.28) has been integrated

Multidimensional Integrable Systems 225

by a method which represents a generalization of that presented in [293] wherein the 8­method was applied to equation (5.1.29). Ifone chooses different operators D, in (5.3.28)

of the form ..ci + >.M i , where ..ci and M i are first-order differential operators, then

one can construct different multidimensional integrable systems as proposed in [592].

This approach involves the Gauss-Codazzi-Ricci equations descriptive of imbedding of

Riemann spaces in multidimensional Euclidean spaces. It leans upon the fact that,

for special types of imbeddings (isometric imbeddings into Euclidean spaces), these

equations can be represented as commutativity conditions

(i, k = 1, ... , n) (5.3.29)

where Ai(x) are algebra G-valued functions [593]. If one parametrizes Ai(x) by a finite

number of fields cpo(x)(a = 1, .. . , N), then equations (5.3.29) lead to a system of

nonlinear differential equations for the functions CPo(x).

One can construct various such multidimensional systems corresponding to different

types of imbeddings and algebras G. An example is provided by the system [594]

+L wfw'f + ePi+Pj = 0,a

(i # j) (5.3.30)

(i # j # k # i)

where

aWf a pi-p oaPi 0---w.e J_=,ax; 3 8x;

(i # j)

D ..(f p) .-~ _ 81 8Pi _ 81 8Pi'3 , . - 8x i8x; 8Xi 8x; 8x; 8Xi'

This system (5.3.30) may be represented in the form (5.3.29) with

(5.3.31)

where MAB are basic elements of the algebra SO(N) (MAB = -MBA). The indices

i,i, k, l take the values 1, . . . ,n, and the indices A and B take the values 1, . .. ,N (N >n ;::: N - n). The indices a and (3 run from 1 to N - n - 1.

Particular cases of the system (5.3.30) have been known for some time [595]. Certain

properties of such systems were studied in [592, 594].

226 Chapter 5

In conclusion, it is noted that integrability in supersymmetric Yang-Mills theory

and supergravity theory in ten-dimensional space have been discussed in [596, 597].

5.4. Obstacles to multidimensionalization of the inverse spectral transformmethod. I. The Born approximation.

In this and previous chapters, a number of multidimensional nonlinear systems

integrable by the 1ST method have been considered. For the most part, these systems

have involved three independent variables . Even then, to construct nontrivial integrable

equations, it is necessary to introduce major modifications of the 1ST method such as

L-A-B triads, quartet, and general operator representations. The multidimensional

(d ~ 4) integrable systems presented in this chapter are equivalent to the compatibility

conditions for linear systems which contain only first-order derivatives, and all arise

out of the second method of multidimensionalization. The number of multidimensional

nonlinear integrable systems with d ~ 4 is much smaller than the number of integrable

systems in 2+1 dimensions (see, for example, [94, 598]). This will be the subject of the

present section .

Consider a possible class of n-dimensional integrable systems which are equivalent

to the compatibility conditions for the system of two equations

(5.4.1a)

(5.4.1b)

where ax = (OXl'· · · ,oxn ) .

There are at least two major reasons why the class of integrable systems repre­

sentable as the compatibility conditions for the system (5.4.1) is extremely narrow. The

first reason is connected with the necessary compatibility of the Born approximations

for (5.4.1a) with the time-evolution equation (5.4.1b). The second reason concerns the

necessary compatibility of the equations which characterize the inverse problem data

for (5.4.1a) with equation (5.4.1b).

Let us discuss compatibility with the Born approximation. To be specific, consider

the case when £1 is a nonstationary multidimensional Schrodinger operator, that is,

£11/1 = «(18y + b. + u(x, y, t))1/1 = 0, (5.4.2)

n 82

where b. = I: 8 2 , U(X1, •• • , xn , y, t) is a scalar function, and (12 = ±1.i=l Xi

As in the case n = 1 (Section 1.4), we introduce the solutions Ff(x, y) of the

problem (5.4.2) defined by their asymptotic behavior as Iyl --+ 00 according to

- 1-2± iAX+ -A yFA (x, y, t) ---+ e (1

y-±oo(5.4.3)

Multidimensional Integrable Systems 227

where X= pol, ... ,An) (-00 < Ai < 00) is an n-dimensional vector. The scattering

matrix SeX', X, t) is defined in the standard manner via

+ j' - ..., ...Fx (x, u, t) = d"AF>,' (x, y, t)S(A ,A,t) . (5.4.4)

(5.4.5)

The evolution law for the scattering matrix can be found, as usual, from equation

(5.4.1b). Let us consider (5.4.1b) as y -+ -00 and insert for t/J the solution pI at

y -+ -00, given by

F+(x y t)1 = jd"A'eiXl:i+Xl2y/CTS(X' Xt).>. " y_-oo ' ,

This procedure yields

dS(~~ X, t) = (y(X') _ Y(XnS(X' ,X,t) ,

where Y(X) = A(iX, iX2)!u::O'The existence of nontrivial solutions of equation (5.4.5) provides a necessary con­

dition for the existence of nontrivial nonlinear integrable equations associated with the

system (5.4.1).

To analyze the possible existence of nonlinear equations integrable via the problem

(5.4.1), we next consider the case of small potential. For potential u(x,y , t) which is

small in some suitable sense, one has the well-knownexpression for the scattering matrix

in the Born approximation [5991

(5.4.6)

where ft(ij, E, t) = (21T)-(n+l)/2 I d"xdyu(x,y, t)eiX:r+iEy is the Fourier transform of the

potential u(x, y, t).

The formula (5.4.5) is also valid, of course, in the Born approximation. Substitution

of the expression (5.4.6) into (5.4.5) yields

(5.4.7)

where if = X, - Xand qn+1 = Xa - X2 . The function u(if,qn+ 1) depends on the variables

if and qn+1, while, in general, the function A(X') - A(X) depends on the variables X, and

X. For equation (5.4.7) to be self-consistent , that is, for u(ij, qn+d to be a function of

the variables if,qn+l at all t, it is necessary that the function A(X') - A(X) be a function

only of the variables if = X, - Xand qn+l = Xf2 - X2. Therefore, for the existence of a

nontrivial solution for equation (5.4.7), it is necessary that a function w(if,qn+l) should

exist such that

(5.4.8)

228 Chapter 5

The functional condition (5.4.8) is a necessary condition for the compatibility of the

system (5.4.1). Thus, all admissible functions A(X) and, therefore, all admissible oper­

ators ~ of the type (5.4.1b) should satisfy the condition (5.4.8). The properties of the

functional equation(5.4.8) for the function A(X)essentially depend on the dimensionality

of the space.

It is readily verified that, in the one-dimensional case n = 1, the condition (5.4.8)

is always satisfied. Indeed, from the definitions ql = A~ - Al and q2 = A? - A~, one has

(5.4.9)

As a result,

A(AD-A(Ad=A(q2+q~) _A(q2-q?) .2ql 2ql

Hence, the function A(AD-A(Al) is, in fact, a function of the variables ql = A~ -AI, q2 =

A? - A~ for any function A(Al) ' For n = 1, the system (5.4.1) is associated with the

KP equation, and the fact that the condition (5.4.8) is fulfilled for any function A(Al)

agrees with the existence of the infinite KP-hierarchy.

A totally different situation arises in multidimensional space. Thus , for n ~ 2, then

condition (5.4.8) is fulfilled only for linear functions A(X) = L Q;Ai within the class ofi=l

polynomial functions A(X) (see [600]) . The proof of this fact is purely algebraic. Let

us start with the simplest case n = 2. First, we introduce the variable <11 = Ai + Al in

addition to the variables ql = A~ - At, q2 = A~ - A2 , and q3 = A?+ Aq - A~ - A~. If we

express AL AI,A~, A2 in terms of ql, q2, q3 , <11 then

2 •\1 _ q3 + q2 - qtqt1\2 - ,

2q2

(5.4.10)

The sets of variables Ai,AI,A~, A2 and ql, q2, q3, ql are connected by a nondegenerate

transformation and provide different parameterizations of the same four-dimensional

space.

Next, let us express A(X/) - A(X) in terms of the variables q), q2, q3, iiI to obtain

A(X/) - A(X) = A (~. + ~ q3 + q~ - qlql) _ A (~. _ ~ q3 - q~ - qlql)2ql 2ql, 2q2 2ql 2ql, 2q2 .

(5.4.11)

The condition (5.4.8) is equivalent to the requirement that the right-hand side of (5.4.11)

be independent of the variable iiI ' For the linear functions A = QlAl + Q2A2 the right­

hand side of (5.4.11) is Qlql + Q2q2 and is, accordingly, independent of ql. For the

Multidimensional Integrable Systems 229

quadratic function A = aA~ + /3AlA2 + 1'A~ the right-hand side of (5.4.11) depends on

ql through the terms

(5.4.12)A 1(/3 ) A l/3qLaqlql + - - l' q2ql - - -ql·2 2 q2

The quantity (5.4.12) must vanish for all ql and q2. This is possible only in the case

a = /3 = l' = O.

A similar situation occurs for an arbitrary polynomial function A(Al, A2) ' Indeed,

let A(All A2) be a polynomial of order N. The condition that the right-hand side of

(5.4.11) be independent of the variable ql is equivalent to the system of equations

(5.4.13)m=l, .. . ,N:q; (A(A~, A~) - A(All A2)) Iql = 0= 0

where A~, A~, All A2 are given by the formulae (5.4.10). The left-hand side of (5.4.13) is

the polynomial L c.; m2m3 q"{'l q;;'2 q';'3 . Since all the powers of qt , q2 , q3 areml+m2+m3=m

independent, equation (5.4.13) at the fixed m is, really, a system of the ~(m+ l)(m+ 2)equations. Thus, for N > 1, the number of equations for the coefficients of the function

A(Al, A2) exceeds the number N + 1 of these coefficients. Consequently, the conditions

(5.4.13) are only satisfied for linear functions A(At ,>'2) '

Accordingly, it is seen that the functional condition (5.4.8) imposes very strong

restrictions on the function A(X) in the multidimensional case. Only linear func­

tions A(X) are admissible within the class of the polynomial functions. As a result,

in the multidimensional spaces (n ~ 2) only first-order differential operators £2 =n

8t + L ai8:z:, + v are admissible [600] . Further, it may be readily verified, that thei=l

compatibility conditions for the system (5.4.1) with such an operator £2 gives rise to atrivial linear equation for u. Hence, we see that consideration of the Born approximation

reveals strong restrictions on the structure of the compatibility conditions.

Analogous restrictions arise for other multidimensional auxiliary systems . Let us

consider the following linear system [277]

£1'" ~ (0, + t.A<0', + P(X,y,t») '" ~ 0,

L2'l/J = (8t + !(8:z:))'l/J = 0,

(5.4.14a)

(5.4.14b)

where AI, .. . ,An are N x N diagonal matrices with distinct elements, ~i = 0 (i =

1, ... , N), N ~ n, and P(x, y, t) --+ O. The operator Ll given by (5.4.14a) is, inx,tI-+oo

fact, the operator £1 in (1.3.36). We introduce the solutions F"1(x , y , t) of the problem

(5.4.14a) with the following asymptotic behavior

(5.4.15)

230 Chapter 5

(5.4.16)

where A = (At, . . . ,An) is an n-dimensional vector. The exponent in (5.4.15) is thesolution of equation (5.1.14a) with P = O. The scattering matrix S is defined via

Ft(x, u, t) = Jd"A'F;'(x, y, t)S(A', A; t) .

It follows from (5.4.14b) and (5.4.16) that the temporal evolution of the scattering

matrix is given by the linear equation

dS(A~~ A, t) = Y(A')S(A', A, t) - S(A', A, t)Y(A) , (5.4.17)

where Y(A) = F(-iX)lp=o is the diagonal matrix. Further, equation (5.4.14a) impliesthe following expression for the scattering matrix S :

S(A', A, t) = O(A' - A) - Jd" x ei I::=t >'~(Amy-xm) P(x, y, t)Ft (x,u, t). (5.4.18)

For small potential P, one has

that is,

(5.4.19)

where FO {3 (qo{3 ,qt, . .. ,qn) is the Fourier transform of the potential PO{3(y, Xl, " . , xn)and

n

qo{3 = L «Am)ooA~ - (Am){3{3Am),m=l

(k = 1, .. . ,n).

The formula (5.4.19) represents the analog of the Born approximation for the matrixproblem (5.4.14a).

Let us now substitute the expression (5.4.19) into equation (5.4.17) to obtain

(o:,/3=l, ... ,N)(5.4.20)

where Yo{3(A) = Oo{3Y2(A). The selfconsistency of equation (5.4.20) requires that the

functions Yo (A')- Y{3(A) should be functions only of the variables qo{3, ql, . . . , qn' There­

fore, the Yo(A) must satisfy the functional conditions

(0'.,/3 = 1,... ,N) (5.4.21)

Multidimensional Integrable Systems 231

where Ba{3 (qa{3 , qi, ... , qn) are appropriate functions. Equations (5.4.21) represent mul-

ticomponent generalizations of equation (5.4.8). As with (5.4.8), the conditions (5.4.21)

give rise to strong restrictions on the admissible form of the functions Ya(A). The

reasoning is completely analogous to that of the previous example.

For n = 1, the conditions (5.4.21) are fulfilled for any function Ya(A). For n 2= 2,

we introduce the n - 1 variables qi = A~ + Ai (i = 1, .. . , n - 1) in addition to the

variables qa{3 , ql, . . . , qn' The set of 2n variables qa{3, ql, ... , qn, ql, . . . ,qn-l is connected

to the set of variables Ai , . .. , A~, A1, . • • , An by a nondegenerate transformation and

gives another parameterization of the same 2n-dimensional space. Let us express

the variables AL· .. , A~, AI, ... , An in terms of the variables qa{3, ql, ... ,qn, ql, . . . , qn- l

and represent Ya(N) - Y{3(A) as functions of the variables qa{3 ,q\, oo.,qn,ql,oo .,qn-l.

The condition (5.4.21) represents the requirement that Ya - Y{3 be independent of the

variables ql, ... ,qn-l . As in the previous example, it may be shown that this conditionn

is satisfied only for linear functions Ya(A) = L Wa(m)Am. In this case, the quantitiesm=l

Wa(m) must obey the following condition [2771

Wa(l) - W{3(l)

aa(l)-al1(l)

Wa(2) - w{3(2)

a",(2) - ap(2)

_W~a~(n..!.)_-_w.::..{3~(n..!.):= °a{3,

a",(n) - ap(n)(5.4.22)

Hence, consideration of the Born approximation leads to the conclusion that only

first-order differential operators £2 are admissible for the problem (5.4.14). In other

words, the linear system (4.3.36) is the only admissible linear system of the type (5.4.14)

in the multidimensional case. As a result, the nonlinear system (1.3.35) is the only

multidimensional integrable system associated with the auxiliary system (5.4.14). We

conclude that the functional conditions of the type (5.4.8) and (5.4.21), which arise out

of consideration of the Born approximation, provide formidable obstacles of a general

character to the multidimensionalization of the 1ST method.

232 Chapter 5

5.5. Obstacles to multidimensionalization of the inverse spectral transform

method. II. Nonlinear characterization of the inverse scattering data.

Another critical impediment to the multidimensionalization of the 1ST method is

revealed by the analysis of the inverse scattering problem for multidimensional linear

problems. These problems have been, and still are, the object of intensive investigation.

Special attention has been paid to the case of the three-dimensional Schrodinger equa­

tion which is very important in certain physical applications. The inverse scattering

problem for spherically symmetric potentials was completely solved some fifty years ago

(see, for example, [262, 601-604]). The generic three-dimensional potential case has

proved to be much more complicated. Pioneering investigations of the general three­

dimensional inverse scattering problem for the Schrodinger equation were undertaken

by Faddeev [283, 285, 604] and Newton [383, 605, 608].

The problem of characterization of the inverse scattering data represents one of

the most important aspects of multidimensional inverse scattering problems [262, 283,

383]. Thus, the usual scattering data in the multidimensional case (d 2: 3) are strongly

overdetermined. For instance, for the problem (5.4.2), the scattering matrix S(>", X)depends on the 2d variables Ai,.. . ,A~, Al,. . . ,Ad, while the potential u(x, y) depends

only on the d + 1 variables Xl , .•. ,Xd, y. In the case d = 1, the scattering matrix, the

inverse scattering data, and the potential all depend on two variables. For this reason,

the characterization problem does not arise. However, for d 2: 2, in order to have

univalent inverse problem equations, it is necessary to impose restrictions on the inverse

problem data which extract functions effectively of d + 1 variables [283, 583, 605].

Important progress in multidimensional inverse scattering problems has been made

possible via the a-method. The expression of the non-analytic dependence of the

solutions in the form of a a-equation has allowed Ablowitz and Nachman [601] to

simplify Faddeev's procedure for the reconstruction of the potential via the scattering

amplitude. The a-approach to the multidimensional inverse scattering problems has

been developed in [291, 292, 609, 610, 611-619]. In particular, the use of the a­approach along with the theory of functions of several complex variables have al­

lowed extension of established results for the three-dimensional stationary Schrodinger

equation [615, 617]. The multidimensional matrix problem (5.4.14a) has also been

studied in detail [610, 612-614]. The penetrating treatment of the characterization

problem for the inverse problem data represents an important achievement of the a­approach. The use of the a-equation has allowed derivation of the system of equations

which govern the inverse problem data. This system is nonlinear [292, 609, 610, 617].

To be concrete, we consider the linear problem (5.4.2), that is,

(aay + fj. + u(x, y))'l/J = 0, (5.5.1)

Multidimensional Integrable Systems 233

where x = (XI, ... ,Xd), a = aR + ia; is a complex parameter, and the scalar po­

tential u(x, y) decreases rap idly as x , y -> 00 [292, 609]. The spectral parameter

A = (A1, . . . , Ad) is introduced in the same way as in the case d = 1 (Section 2.1),

namely, by transition to the function X(x, y, A) := 'l/J (x, y) exp ( iAX+ A:y) , AX :=

AIXI + ...+ AdXd, where A = AR+ iA/, AR < O. The function X obeys the equation

a8y X+ ~X + 2iAVX + ux = O. (5.5.2)

We consider the solutions of equation (5.5.2) bounded for all x , y and canonically

normalized in A (X --+ 1). Such solutions satisfy the integral equation1.\1-+00

X(X, y) = 1- (GuX)(x ,y) , (5.5.3)

where (Gcp)(x,y) := J Jdnx'dyG(x-x',y-y';A)Cp(X' ,y') and G is a bounded Green

function for the operator Lo = a8y + ~ + 2i'XV. A straightforward calculat ion gives

JJe i(ii(+Yl7)

G(X,y;AR,A/) = (27l")-d-1 ~E;,dT/. ~ __zaT/ - E;, - 2AE;,

= Sg: (y) (27l")-d J~E;,e!;(f+2X()+ix((}(_yaR (f + 2 (An + ~:/) ~ . (5.5.4)

The Green function (5.5.4) is nowhere analytic. Consequently, the solut ion X of equation

(5.5.3) is likewise nowhere analytic. As in the case of the KP-II equatio n (d = 1), we

must calculate :~ = ~ (8~~i + i :.x~i)· To do this, we first differentiate (5.5.3) with

respect to ~i to get

:~ = - ;fux - G ( u :~) ,

and on use of (5.5.4), we obtain

(~~ U X) (x ,y)

= (27l")~laRI J~E;,ei/3(x ,y'.\R.At ,e)T(AR,A/ ,E;,)(E;,i - AR.)8(S(E;,)) ,

where

(5.5.5)

(5.5.6)

(5.5.7)

234 Chapter 5

Accordingly, a'!- obeys the integral equation (5.5.5) with a free term of the forme})\i

(5.5.6). In view of this, it is natural to introduce the function It which satisfies the

integral equation

(5.5.8)

whence, we obtain

(5.5.9)

As in the case of the KP-II equation, it is now necessary to determine the relation

between the functions It and x. It follows from (5.5.4) that the Green function G

possesses the symmetry property

(5.5.10)

on the surface S(e) = O. Comparison of (5.5.3) with (5.5.8), in view of (5.5.10), yields

(5.5.11)

axaXi =

(5.5.13)

on S(e) = O. Substitution of (5.5.1) into (5.5.9) then gives [292, 609]

(21r);laR) II d"e(ei - ARI )c5(s(e) )ei{j(X,Y~>'R '>'1 ,e)T(AR, AI,e)x(x,y, e, AI),

(i = 1, ... ,d).(5.5.12)

Thus, for the bounded solutions of the problem (5.5.2) we obtain a special nonlocal

a-problem, or more precisely, the system of d nonlocal a-equations (5.5.12).

The a-system (5.5.12) generates the inverse problem equations. These are given by

a multidimensional generalization of the formula (1.4.10) and adopt the form [292, 609]

1II ' ,*t(X,y,A'n,Aj)X(X,y,AR,AI) = 1+;: dAR;dAI;' Ai _ A~ ,

where :r are given by the right-hand side of (5.5.12), A'n; := (AR I , · · · , A'n;, . . . , ARJ

and similarly for Aj; . The multidimensional singular integral equation (5.5.13) together

with the reconstruction formula

(5.5.14)

which follows directly from (5.5.2) and (5.5.13) gives the solution of the inverse problem

for the linear equation (5.5.2). The function T(AR, AI,e) on S(e) = 0 provides the

inverse problem data.

Multidimensional Integrable Systems 235

In the simplest case d = 1, the system (5.5.12) reduces to the single quasilocal

a-equation [292)

:~ = 211"1~RI sgn (AR + ;~ AI)ei.B(x,!I,>'R,>./,eo)T(AR, AI,~o)x(x, u, ~o, AI), (5.5.15)

where A == Al and ~o = -AR - 2 (11 AI. It may be verified that, in the cases (11 =(1R

0, (1R = -1 and (11 = 1, (1R -+ 0 (with the redefinition ~ = ~), the a~uation (5.5.15)(1R

coincides with the a-equations for the KP-II equations respectively. In the case d = 1,

the function T(AR, AI,~o(AR, Ad) is a function of two variables and the possibility of

overdetermined inverse problem data does not occur.

In the multidimensional case d ~ 2, the function T(AR,AI ,~) on the subspace

S(~) = 0 is a function of 3d - 1 variables, while the potential u(x, y) depends on d + 1

variables. For this reason, to make the forward inverse scattering problems uniquely

solvable it is necessary to impose certain restrictions on T(AR, AI, ~).

Another feature of the multidimensional problem (5.5.2) is that, instead of a single

a-equation (5.5.15), we have a system of a-equations (5.5.12). The question of the

compatibility of this system arises. In fact, the compatibility conditions for the system

(5.5.12) consist, of course, of the system of conditions 882

8x = 8~2: ' (i =J k).

Ai Ak Ak AiThe important observation was made in [609) that these are just the conditions which

provide the explicit characterization equations for the inverse problem data. These are

of the form [292, 609)

where

and

£.ijT(AR' AI,~) = Jcr~'~ o(S(~'))T(AR, A/,~')T(~', AI, ~),'J

~ = (211")~I(1RI {({j - AjR)(~i - ~D - (~~ - AiR)(~j - ~j)} .'3

(5.5.16)

Thus, in order for the function T( AR, AI,~) to represent the inverse problem data for

some potential u(x,y), it must obey the system of equations (5.5.16).

The system (5.5.16) is quadratic in T. This leads to an important negative result if

one tries to associate with (5.5.1) the nonlinear integrable equation which is equivalent

to the compatibility condition for the system (5.4.1). Thus, equation (5.4.1b) dictates

that according to the inverse problem data, T(AR' AI,~) should evolve

236 Chapter 5

where the form of the function 11 is defined by the form of the operator A. However,

it may be checked that such time-dependence for T(AR,)'/, t) with nontrivial function

11 is not admissible by the nonlinear characterization equations (5.5.16). Thus, the

nonlinearity of the characterization equations (5.5.16) for the inverse problem data

at d ~ 2 is a major impediment to the construction of nontrivial multidimensional

integrable equations associated with the linear system (5.5.1) [292] . Nonlinear charac­

terization equations of quadratic integrodifferential type for the inverse scattering data

also arise for other multidimensional scattering problems associated , for instance, with

the stationary Schrodinger equation and the matrix problem (5.4.14a) (see [292, 610,

618]). Again, this nonlinearity is an obstacle to multidimensionalization. However,

the special matrix structure of the problem (5.4.14a) provides a special situation. In

that case, the corresponding characterization equation turns out to be compatible with

evolution of the inverse scattering data in the form

T(A', A, t) = eY(A')tT(A', A, O)e-Y(A)t,

where Y(A) is a diagonal matrix-valued function which is linear in Ai (i = 1, .. . , d)

[292, 610]. The corresponding nonlinear integrable system (1.3.35) is formally (d +I)-dimensional but is actually only three-dimensional. Thus , by using the special

structure of the quantities 11",.8 (5.4.22), Manakov (unpublished) and Fokas [613] have

demonstrated that the system (1.3.35) is locally reducible to one which is spatially

two-dimensional by the introduction of characteristic coordinates .

To summarize, in the multidimensional case (d ~ 2) there exist serious imped­

iments to the construction of certain types of multidimensional nonlinear integrable

systems, at least by standard means. Methods to overcome these obstacles remain to

be developed. One possibility being currently discussed involvestransition from theories

described by point quantities (functions of a finite number of coordinates) to theories

of extended objects such as strings, surfaces, etc. [462, 620, 621] . In this connection,

attempts to construct exactly solvable multidimensional models which develop ideas

due to Zamolodchikov [620, 621] have been undertaken recently in [332, 622].

Conclusion

In this volume, we have restricted ourselves to methods of construction of multi­dimensional nonlinear integrable equations via the inverse spectral transform method .

We have not discussed various remarkable properties which these equations enjoy such

as associated conservation laws, symmetry groups, Backlund transformations, Hamilto­

nian, and recursion structures. The reasons for this are twofold. On the one hand, their

scope demands an independent study. On the other hand , the algebraic properties ofmultidimensional integrable equations have turned out be be much more complicated

than in the (1 + I)-dimensional case, and there remains much to be understood. Nev­

ertheless, to conclude, we indicate certain important papers in these areas so that the

interested reader may consult them directly.

Conservation laws for multidimensional integrable equations have been discussed

in [94, 319, 412, 457-460, 623-628] . In the (2+ I)-dimensional case, the formal integralsof motion can be true constants of motion or constraints, depending on the class of

the solutions under consideration [625] . Integrals of motion in formal series form,

degenerate dispersion laws, classical scattering matrices, and integrability criteria in

multidimensional spaces have been studied in [261, 319, 626, 627, 629].

Infinite-dimensional symmetry groups of multidimensional integrable equationshave been considered in [630-648] . Both commuting and nonabelian symmetries with

explicit dependence on the coordinates x , y, t, . . . are among these symmetries.

Backlund transformations have been studied in a number of papers [344, 413,649-666] . Various classes of exact solutions of different (2 + I)-dimensional integrableequations have been constructed by means of Backlund transformations [268, 344, 413,

649, 650, 659, 662, 663]. In a recent very important development, new interesting

solutions of the DS equation (and indeed the whole DS hierarchy) which decrease

exponentially in all spatial directions have been constructed in [665, 666] by means of

elementary Backlund transformations [662]. This has led to the new subject of dromion

theory.

The Hamiltonian structure of multidimensional integrable equations has been in­

vestigated in [196, 319, 667-679] . The possible generalization of the notion of classical

237

238 Conclusion

r-matrix to the {2+ I)-dimensional case has been discussed for the KP and DS equations

in [317, 332, 335, 673-678].

The recursion structure of {2+ I)-dimensional integrable equations has been in­

vestigated in some detail in the literature. The local approach was first proposed in

[680-682] (see also [118, 684)) . It was proved in [243] that the usual local recursion

operator does not exist for nonlinear (d + I)-dimensional Hamiltonian equations for

d ~ 2. The bilocal approach was introduced in [440, 685]. Important progress was made

in [686-689] where it was shown that the KP hierarchy can be written in compact form

via the bilocal recursion operator. The bilocal approach was subsequently developed in

a series of papers [689-710]. In this bilocal formalism, the KP hierarchy of equations is

representable as

n = 1,2,3, (6.1)

where an are constants, tl is the projection operator tl¢{x, z', y, t) := ¢{x, x', y , t)lx'=x,

and the operator A is given by

A = ~{8x' + 8x )- 1{a8y + 8;, - 8; + u(x', y) - u(x, y)} . (6.2)

The form (6.1) of the KP hierarchy and the bilocal recursion operator (6.2) are analogs

of the corresponding form (1.2.3) and the recursion operator A as given by (1.2.32) for

the KdV hierarchy. The basic feature in the {2 + I)-dimensional case is the bilocal

character of the recursion operator (6.2). The analog of the formula (1.2.33) adopts theform [708]

(AX('\)X('\)) (x' , x) = '\X(x', y, t)x(x, y, t), (6.3)

where X{x' , y, t) satisfies equation (1.4.4a), while X{x , y, t) satisfies the adjoint equation

(-a8y + 8; - 2i'\8x + u(x, y, t))X(x , y, t) = O.

Bilocality is also associated with the r-function approach (Section 4.1). Indeed, bilocal­

ity is a fundamental feature inherent to the structure of the multidimensional integrable

equations.

In conclusion, we mention recent papers [711-745] devoted to diverse aspects of

multidimensional integrable equations. We note among them the series of papers

[714-716, 725, 727] which involve algebraic formulations of the a-dressing method and

the paper [744] in which the initial-boundary value problem for the DS-II equation isconsidered.

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Adjoint eigenfunction, 159Adjoint problem, 159

Benney system, 154Bilinear identity, 160

Cylindrical Kadomtsev-Petviashvili(Johnson) equation, 132, 133

Davey-Stewartson equation, 23, 76-90DBAR-equation (problem) , 41, 44, 56,

95, 102, 172-192

Index

Lax pair, 3Lump, 35, 67, 68, 83, 85, 99, 100, 120,

121, 126, 127

Manakov triad, 27Manakov-Zakharov-Mikhailov equation,

26, 121-130Melnikov system, 134-139Modified Kadomtsev-Petviashvili

equation, 22, 140Modified Nizhnik-Veselov-Novikov

equation, 147

Fredholm alternative (theorems), 50, 51Fredholm determinate, 50, 61, 81Fredholm integral equation, 42

Gardner equation in 2+1 dimensions, 143Generalized (pseudo) an alytic function,

44, 45, 63-67, 67Generalized Cauchy formula, 41Green function , 16, 48-53, 59-61, 78-80,

87, 92-94, 101, 104, 107, 123, 124

Harry Dyrn equation in 2+1 dimension,145

Hirota method, 118-120, 163

Initial value (Cauchy) problem, 2, 42, 57,6~ 85, 89, 98, 10~ 110, 126

Ishimori equation 24, 116-121

Kadomtsev-Petviashvili equation, 22,47-76

Kaup-Kuperschmidt equation in 2+1dimensions, 144

291

Nizhnik equation, 28, 106-110Nonabelian Radon transform, 125Nonlocal Kadomtsev-Petviashvili

equation, 130-133

Operator form of integrable equations, 6,10, 27, 30, 32, 192-202, 218

Riemann-Hilbert problemlocal , 8, 13-18, 43, 125, 128nonlocal, 38-40, 43, 54, 55, 89, 95,

109

Sawada-Kotera equation in 2+1dimensions, 144

Scattering matrix, 36Self-dual Yang-Mills equation, 33,

203-212in multidimensions, 224, 225

Sine-Gordon system in multidimensions,219-223

292

Singular integral equation, 62Solutions with functional parameters, 39,

71, 72, 86, 180, 181Supersymmetric Yang-Mills equation,

213-218

Tau function, 161-166

Index

Vertex operator, 156-167Veselov-Novikov equation, 28, 91-100

Wave equation in multidimensions,219-223

Zakharov-Manakov system, 32