30 years of ﬁnite-gap integration...

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30 years of finite-gap integration theory

BY VLADIMIR B. MATVEEV*

Universite de Bourgogne, Institut de Mathematique de Bourgogne,21078 Dijon Cedex, France

Themethod offinite-gap integration was created to solve the periodic KdV initial problem.Its development during last 30 years, combining the spectral theory of differential anddifference operators with periodic coefficients, the algebraic geometry of compact Riemannsurfaces and their Jacobians, the Riemann theta functions and inverse problems, had astrong impact on the evolution ofmodernmathematics and theoretical physics. This articleexplains some of the principal historical points in the creation of this method during theperiod 1973–1976, and briefly comments on its evolution during the last 30 years.

Keywords: algebraic curves; solitons; integrable systems; theta functions;finite-gap operators; Abelian integrals

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1. Introduction

Afterbeing invitedbyVadimtowrite the introductoryarticle to this volume I startedto hesitate. A question was: what should it be about? It is certainly impossible tocover thewhole field taking into account thousands of publications and several bookscompletely (Belokolos et al. 1994; Gesztesy &Holden 2003) or partially (Marchenko1977;Wilson 1981;Mumford 1983; Novikov et al. 1984; Bogoyavlenskij 1991; Dickey1991; Cherednik 1996; Kaku 1998) devoted to this topic. Finally, I have decided todiscuss in more detail the very first developments of the theory and to sketch brieflysome of the most important developments of the last 30 years.

(a ) Inverse scattering method before the finite-gap integration

After the fundamental discovery of the inverse scattering method made byGardner et al. (1967), allowing a solution of theCauchy initial problemwith rapidlydecaying initial data for the KdV equation, there was a short delay before theappearance of seminal works by Lax (1968), Zakharov & Shabat (1971, 1972) andAblowitz et al. (1973). In theseworks, itwas realized that the samemethodmight beapplied to an infinite number of 1C1 field theoretic models, and that among thesethere were several models of fundamental importance for the physics of nonlinearphenomena. Soon after, the soliton virus started to propagate across the world.Many importantmodels were solved, in the spirit of the above-mentionedworks, byManakov, Zakharov, Shabat, Faddeev, Takhtajan, Ablowitz, Kaup, Newell and

Phil. Trans. R. Soc. A (2008) 366, 837–875

doi:10.1098/rsta.2007.2055

Published online 26 June 2007

ish to dedicate this article to the bright memory of Vadim Kuznetzov.

e contribution of 15 to a Theme Issue ‘30 years of finite-gap integration’.

[email protected]

837 This journal is q 2007 The Royal Society

http://rsta.royalsocietypublishing.org/

V. B. Matveev838


Segur during the period 1973–1974, and the first solitonic teams were formed in theformer Soviet Union, the USA, Italy and other countries.

The finite-gap integration method was created in 1974–1976 in the formerSoviet Union and the USA. In the next few subsections the development of themethod is presented in a narrative form and as such concerns mostly the ‘Russianside’ of the story,1 but even in this part it is very incomplete. It reflects mainlythe author’s personal recollections in the belief that it might be of some interestfor new generations of researchers.

(b ) Ufa 1973: ‘how to solve the KdV periodic initial problem?’

In October of 1973, the first soliton conference in the Soviet Union bringingtogether 30 participants was organized in Ufa. Among the participants wereseveral people who had already made some crucial contributions into the field ofintegrable models: Zakharov, Shabat, Faddeev, Manakov and Takhtajan. Therewere also many people (including the present author) fascinated by the newpromising area of activity, but at that moment still having made nocontributions in the ‘solitonic sector’. In particular, Arnold, Kirillov, Maninand Novikov were among the participants.

At that conference many new impressive results were first presented, includingthe solution of the Toda lattice model and the vector NLS model by Manakov,the solution of the sine-Gordon model by Zakharov, Faddeev and Takhtajan,with strong emphasis on the Hamiltonian interpretation and the construction ofthe related action-angle variables.

One of the topics widely discussed in the couloirs of the conference was how tosolve the KdV-like nonlinear equations for periodic initial data and, in particular,what kind of animals the periodic analogues of solitons should be.

To understand the difficulty of the problem and its distinction from the case ofthe rapidly decaying initial data, it is necessary to emphasize that the basictechnical tool for solving KdV with rapidly decaying initial data was the solutionof the highly non-trivial inverse scattering problem for the Schrodinger operatoron the line obtained by Faddeev 10 years before the appearance of the article byGardner et al. (1967).2 The so-called reflectionless potentials taken as initial datafor the KdV equation were generating the famous multi-solitons solutions. Thepossibility of describing the multi-soliton solutions by explicit determinantformulae was the direct consequence of the degeneracy of the kernel of the relatedMarchenko Fredholm-type integral equations.

The efficient solution of the inverse spectral problem for the Schrodingeroperator with periodic potential at that time was unknown. The only knownperiodic solution of the KdV equation was the simple cnoidal wave solutionconstructed almost 100 years previously by Korteweg and de Vries.

Tentative answers to the question of how to solve the KdV initial problemcirculating around the conference discussions appeared later to be misleading.
1 Parallel developments in the USA started in the works of Kac & van Moerbeke (1975), Lax(1975), McKean & van Moerbeke (1975), Flaschka & McLaughlin (1976).2 Faddeev’s solution was based on previous works by Gelfand, Levitan, and especially byMarchenko, on the inverse scattering problem on the half-line (see Marchenko (1977) for moredetailed references).
Phil. Trans. R. Soc. A (2008)




(c ) Akhiezer’s work, first breakthroughs: Novikov’s equations, trace formulaeand moving Dirichlet eigenvalues

This subsection describes what happened after the Ufa conference during theperiod between November 1973 and January 1974.

Coming back from the Ufa conference to St Petersburg, I started to thinkabout the periodic KdV problem. I was at the time probably the only one to sensethe importance of the work of Akhiezer (1961). I learned about the work ofAkhiezer in 1964, being a fourth year student of the physical faculty of Leningrad(now St Petersburg) University from the excellent book of Glazman (1963). Inhis work, Akhiezer claimed the existence of multi-parametric families of periodicSchrodinger operators with a finite number of spectral gaps. Even more, hereduced the reconstruction of some subfamily of the finite-gap potentials to theclassical Jacobi inversion problem on hyperelliptic Riemann surfaces.

While coming back from the Ufa conference, some 10 years after taking a firstlook at Akhiezer’s article, I got the strong feeling that his work, after being properlyunderstood, should be the key for solving the KdV periodic Cauchy initial problem.

In general, the spectrum of the Schrodinger operator with a periodic potentialcontains an infinite number of closed intervals separated bygaps. In particular, this isthe case of the Kronig–Penny model with ‘rectangular’ periodic potentials. TheKronig–Penny model was the only solvable periodic model then known in themajority of the solid-state physical community since the creation of quantummechanics.Therefore, Iwas rather surprisedbyAkhiezer’s result. InAkhiezer’swork,the existence of the class of Schrodinger operators having an absolutely continuousspectrum consisting of a finite number of intervals separated by gaps was establishedvia the inverseproblemapproach.Moreprecisely,Akhiezer succeeded in reducing thereconstruction of the potential from some special class of spectral functions to thesolution of the Jacobi inversion problemon the hyperelliptic Riemann surface, whosebranch points were coincident with the boundaries of the gaps. The solution of theJacobi problemwas not achievedbyAkhiezer; he presented thefinal result only in theone-gap case, corresponding to the simplest Lame elliptic potential.

It was very inspiring, due to a known fact that one of the infinite period limitcases of the simplest Lame potential was the reflectionless potential with onediscrete negative eigenvalue.

In general, the KdV flow of the simplest Lame potential coincided with thesimple cnoidal wave solution, found long ago by Korteweg and de Vries—the onlyx -periodic solution of the KdV equation known for almost a century. It was alsowell known that one of the infinite period limit cases of this solution coincided withthe one-soliton solution of the KdV equation. Therefore, the idea that the finite-gap potentials might represent the kind of initial data for which we could expect tocalculate the solution of the Cauchy initial problem explicitly became natural.

Therefore, at that moment, the question of how to attack the periodic KdVproblem was reduced to the following more technical questions.

(i) How does one characterize and construct explicitly all smooth, real valuedperiodic finite-gap potentials?

(ii) What is their KdV dynamics?(iii) Taking into account the fact that the Schrodinger equation with the simplest

Lame potential was explicitly solvable in elliptic functions, there was also a



V. B. Matveev840


natural question: is it possible to solve explicitly the Schrodinger equationwith the finite-gap potential?

It is necessary to say that at that time, elliptic functions, theta functions, Abelianfunctions and integrals were almost excluded from the standard curricula ofmathematics and especially of physics faculties of the majority of universities allover the world. Therefore, to understandAkhiezer’s work better than he did, it wasnecessary to learn these things. Many books on Abelian functions written byrecognized mathematicians (a typical example is the book of C. Chevalley onalgebraic functions) were not adapted for quickly introducing the potential readerinto the subject.

Therefore, instead of dedicating all my time in this direction, I started to thinkabout an alternative approach of a ‘dynamical nature’. It was clear from thespectral theory (Titchmarsh 1958; Glazman 1963; Magnus & Winkler 1979) ofthe one-dimensional periodic Schrodinger operators H

H ZKd2

dx2CuðxÞ; uðxCTÞZ uðxÞ; ð1:1Þ

that the spectra associated with different boundary value Sturm–Liouvilleproblems behaved differently with respect to the deformations of the potentialgenerated by simple translations of its argument or by the KdV flow. It wasobvious that the boundaries of gaps Ej coinciding with eigenvalues of periodic oranti-periodic Sturm–Liouville problems

Hy Z ly; ð1:2Þ

yð0ÞZGyðTÞ; y 0ð0ÞZGy 0ðTÞ; ð1:3Þwere the integrals of motion: they are the same for initial potential u(x), for itstranslations u(xCt) or its deformations by the KdV flow u(x, t),

ut Z 6uu xK u xxx ; uðx; 0ÞZ uðxÞ: ð1:4ÞBy contrast, the eigenvalues lj of the Dirichlet boundary value problem

Hy Z ly; yð0ÞZ yðTÞZ 0; ð1:5Þshould move inside the gaps [E 2j,E 2jC1] acquiring dependence on t and t. Thismotion is periodic with respect to the translation parameter t. The minimal andmaximal values of lj (t), considered as a function of t, coincide with lower andupper boundaries of the j -th gap

min ljðtÞZE2j ; max ljðtÞZE2jC1:

This picture concerns any real smooth periodic potential. It follows immediatelyfrom Lyapunov inequalities, discovered approximately 100 years ago (Glazman1963), showing that in the case of the continuous periodic potential there isexactly one Dirichlet eigenvalue inside the closure of any spectral gap.3

Now a question is, what was so special from the point of view of this pictureconcerning the finite-gap case? It was clear that in the finite-gap case only the
3 In Akhiezer’s original construction (Akhiezer 1961), the related lj were assumed to coincide withboundaries of spectral gaps. The fact that the generic choice of these boundaries leads to the almostperiodic potentials (first time mentioned by Novikov (1974) in a context of discussion of the finite-gap solutions of the KdV equation) was also not observed by Akhiezer.




Dirichlet eigenvalues lying in the closure of the non-degenerate gaps coulddepend on the translation parameter t or on the KdV time variable t: when thegap is contracted to the point, i.e. E2jZE2jC1 there is no room to move inside it.At this moment I asked myself, what will happen if we take the difference of thefamous Gelfand–Levitan–Dickey trace identities (Dickey 1955), written forthe Dirichlet eigenvalues of H, with the translated and non-translated potential?The result for the first three trace identities (valid for any smooth periodicpotential) looks as follows (after subtraction we replace t by x):

XNjZ1

ðljðxÞKljð0ÞÞZKuðtÞ

2

��x0

; ð1:6Þ

XNjZ1

l2j ðxÞKl2j ð0Þ� �

ZK2u2ðtÞCu 00ðtÞ

4

��x0

; ð1:7Þ

XNjZ1

l3j ðxÞKl3j ð0Þ� �

ZK16u3ðtÞC24uðtÞu 00ðtÞK3uð4ÞðtÞC15u 02ðtÞ

32

��x

0

: ð1:8Þ

In general, for any smooth periodic potential, using the results of Dickey(1955), we obtain the infinite sequence of identities

XNjZ1

lkj ðxÞKlkj ð0Þ� �

ZPk u 0ðtÞ;.; uð2kÞðtÞ� ��x

0; k Z 1; 2;.;

where the polynomials Pm can be computed using the simple recurrence relationsobtained in Dickey (1955).

A slightly more precise form of the trace identities might be obtained in asimilar way, completing the Gelfand–Levitan–Dickey (equal to GLD) formulaeby similar trace identities written for the eigenvalues of periodic and anti-periodic Sturm–Liouville problem. In general, this leads to the infinite series ofidentities linking the Dirichlet eigenvalues with eigenvalues of periodic and anti-periodic Sturm–Liouville problems and the potential, which is assumed to beperiodic and smooth. It is necessary to emphasize that the same formulae remainvalid if we replace u(x) by u(x, t), where u(x, t) is the solution of the KdVequation with initial data u(x, 0)Zu(x). The only difference is that the Dirichleteigenvalues acquire the time dependence ljZlj(x, t). In the g-gap case, assumingthat exactly first g-gaps are not closed, the first three of those identities are

XgjZ1

ljðx; tÞZKuðx; tÞC

P2gC1

k

Ek

2; ð1:9Þ

XgjZ1

l2j ðx; tÞZKu2ðx; tÞC

P2gC1

k

E 2k

2C

u 00ðx; tÞ4

; ð1:10Þ



V. B. Matveev842


XgjZ1

l3j ðx; tÞZKu3ðx; tÞC

P2gC1

k

E 3k

2C

3

4uðx; tÞu 00ðx; tÞK 3

32uð4Þðx; tÞC 15

32u 02ðx; tÞ;

ð1:11Þwhere prime denotes differentiation by x.

Therefore, in order to reconstruct the finite-gap potential with given values ofboundaries of non-degenerate gaps (or, respectively, the solution of KdVequation) with finite-gap initial data, it is enough to find the sum of movingDirichlet eigenvalues.

In the absence of gaps in the spectrum of H, the first of the above identitiessays that the potential u(x) is constant: u(x)Zu(0).

In the one-gap case, assuming that only the m-th gap is not closed, weconclude that the motion of the m-th Dirichlet eigenvalue determines thepotential up to a constant uðxÞZK2lmðxÞCumð0ÞC2lmð0Þ. In the same one-gapcase, excluding lj(x) from the identity (1.7), and differentiating once, we see thatany one-gap periodic potential represents the solution of the generalizedstationary KdV equation

6uuxK uxxxKvux Z 0; v ZK4ðE1CE2CE3Þ: ð1:12ÞTherefore, at this point it becomes obvious that any one-gap periodic potentialhas the form uZ2PðxÞCc, where cZKv/4cZKv=4 is an arbitrary constantand P is a Weierstrass elliptic function. In particular case when E1CE2CE3Z0,the one-gap potential represents the solution of the stationary KdV equation. Itis clear that, in general, the one-gap potential generates the simple wavesolution u(xKvt) of the KdV equation, so that the boundaries Ej, jZ1, 2, 3 ofnon-degenerate gaps determine the velocity of its propagation, according toequation (1.12).

Quite similarly, in the finite-gap case, assuming that only g first gaps4 are notclosed we get from (1.6) the formula

uðtÞZK2XgjZ1

ljðtÞCuð0ÞC2XgkZ1

ljð0Þ;

showing that the t-dynamics of the Dirichlet eigenvalues, moving inside of non-degenerate gaps, completely determines the potential. It is also clear that theamplitude of oscillations of the potential, as well as the amplitude of the relatedsolution of the KdV equation, does not exceed the sum of the lengths of thegaps. Therefore, for the narrow gaps, the amplitudes of the oscillations of thepotential u(x, 0) and of the related solution u(x, t) of the KdV equation arealways small.

In general, the trace identities mean that any finite g-gap periodic potentialsatisfies some nonlinear ordinary differential equation of the order 2g, whichcan be proved in a same way, excluding the g ‘moving’ Dirichlet eigenvalueslj(x)Ks (belongings to the closures of non-degenerate spectral gaps), from thefirst g trace identities.
4Of course, the same results remain valid in the case when any g-gaps are not closed, with obviousmodifications of enumeration of lj.




I was at this point at the end of 1973,5 when L. D. Faddeev told me at thebeginning of January 1974 that in Moscow S. P. Novikov had discovered theimportance of the finite-gap potentials for solving the periodic KdV problem andwas working actively in this direction. I informed Faddeev about Akhiezer’s workand gave him Akhiezer’s article for Novikov (since he was going to Moscow for afew days) at the middle of January. Faddeev brought back the manuscript of thework of Novikov (1974), which contained many remarkable results, in a sensecomplementary to those I had discovered for myself up to that moment.

First, Novikov came to the idea of importance of the finite-gap potentials whilelooking for the spectral interpretation of the simple wave solution. This solution

u Z 2PðxKvtÞ;where P(x) is a Weierstrass elliptic function, is obtained by the directsubstitution of the anzatz uZu(xKvt) in (1.2) and leads after two quadraturesto a problem of inversion of the elliptic integral, which is naturally solved bymeans of the Weierstrass function. Next, he learnt from Arnold, as he explainedto me later, that there existed a work by Ince (1940a,b), where it was shown (seealso Erdelyi et al. (1955) where the results of Ince are reproduced) thatthe spectrum of the 1D Schrodinger equation with Lame periodic potentialuðxÞZnðnC1ÞPðxÞ contains exactly n gaps. Inspired by this information,combined with the well-known fact that, when the imaginary period tends toinfinity, the simple wave periodic solution goes to the one-soliton solution,Novikov conjectured that the finite-gap potentials should be the natural periodicanalogues of the reflectionless potentials, and their KdV dynamics shouldgenerate the natural extension of the multi-soliton solutions to the periodic case.He posed the same question as me but attacked the problem from a differentdirection. Namely, he proved in his first work (Novikov 1974) the followingstatement. Let Ij be the integrals of motion of the KdV equation corresponding tothe periodic initial problem (see Novikov et al. (1984) for definitions and details).Then, any periodic solution of the higher stationary KdV equation

dIgduðxÞC

XgjZ1

cjdIgKj

duðxÞ Z d ð1:13Þ

represents the g-gap periodic potential.6

Therefore, Novikov’s theorem gave the nonlinear ordinary differentialequations that provided the sufficient conditions for the potential to producethe g gaps in the spectrum, while my own approach, based on the Gelfand–Dickey trace formulae, provided the necessary conditions. Despite the differentform and the different way to derive them, for gZ1, 2, 3, the ordinary differentialequations (coming from completely different points of view) appearing inNovikov’s and my approaches were in fact the same. Therefore, it was natural toconjecture that they should coincide for any value of g, although at that momentthe proof was still missing.
5 Slightly later, I found the article by Hochstadt (1965) where he proved in a similar way a morespecial result, saying that any smooth one-gap potential is expressed by means of the Weierstrass Pfunction. Hochstadt probably ignored the existence of the infinite series of GLD identities stoppinghim at the one-gap level.6 The same statement was proved in a different way by Lax (1975).


V. B. Matveev844


In Novikov (1974), it was also proved that the variety of solutions of equation(1.13) is invariant with respect to the actions of the KdV and higher KdV flows.The action of those flows on the variety of solutions of equation (1.13) iscommutative. Novikov also proved that equation (1.13) represents a completelyintegrable Hamiltonian system with n degrees of freedom depending on nC1parameters (c1,., cg, d ), whereby the collection of g commuting integrals of thissystem and all the parameters (c1, ., cg,d ) are expressed in terms of the 2gC1boundaries of the spectral gaps.

Another important ingredient of Novikov’s work was the study of theevolution of the monodromy matrix of the periodic Schrodinger operator withrespect to both a change of the reference point and the KdV flow. The importantby-product of this study was the discovery of the zero-curvature representationof the KdV equation with 2!2 matrices depending in a polynomial way on thespectral parameter.

Novikov also mentioned that, for generic choices of the constants cj, thesolutions of the higher stationary KdV equations (nowadays called Novikov’sequations) should be almost periodic functions of x, tentatively representing thealmost periodic g-gap potentials. The correct definition of the integrals of motionin the almost periodic case and the proof of their existence, somehow, wasmissing in Novikov (1974), mainly based on spectral theory corresponding to thestrictly periodic case. The spectral theory for this kind of almost periodicproblems was also not yet constructed at that moment.

In Novikov (1974), it was also pointed out that the reflectionless potentials inthis picture correspond to degenerate separatrix solutions of equation (1.13),corresponding to the special choice of the constants cj and d.7

The Novikov (1974) work was extremely important for further development.In his work, the Hamiltonian approach to the finite-gap integration theory wasdeveloped for the first time. It was very inspiring for many further resultsobtained by Novikov himself, and his school, and also by Gelfand and Dickey andmany other researchers.

However, the principal questions, concerning the explicit description of the finite-gap potentials and related solutions of the KdV equation, still remained open.

It became clear soon that the attempts to solve explicitly the highly nonlinearordinary differential equations describing the g-gap potentials (independently ofthe chosen way to generate them) by ‘brute force’ were hopeless and the centre ofgravity of further efforts reverted to further understanding Akhiezer’s work.

(d ) Return to Akhiezer. Solution of the KdV periodic Cauchy problemand theta-functional formulae

At this moment new actors came to the game: Boris Dubrovin (at that time asecond year PhD student of Novikov at the Mathematical Faculty of MoscowUniversity) and Sasha Its (at that time my fifth year student at the Faculty ofPhysics of St Petersburg University). The problem of finding a complete,
7 It is necessary to mention here also the works by Marchenko (1974) and by Marchenko &Ostrovskij (1975), where some prescriptions to construct the boundaries of the gaps, correspondingto the strictly periodic potentials, were given in terms of Schwartz–Christoffel integrals.Marchenko & Ostrovskij (1975) also proved an approximation theorem, establishing the densityof finite-gap periodic potentials in the space of all periodic potentials; see also Marchenko (1977).




efficient, description of the whole class of smooth real-valued g-gap periodicpotentials and related solutions of the KdV equation was completely solvedduring the following three months independently, and with certain importantvariations, by Its and myself in St Petersburg and Dubrovin in Moscow (seeDubrovin 1975; Its & Matveev 1975a).

The main ingredients of the solution obtained by Its and myself were thefollowing. First, using the spectral theory of the general one-dimensional periodicSchrodinger operators, we proved that for any g-gap periodic potential the pair ofBloch solutions j1,2(x, l) of equation (1.2) can be considered as a single functionj(x, P), where PZ(w, l) varies on a hyperelliptic curve G of genus g,

GZ ðw; lÞ : w2 ZQðlÞ; QðlÞ :ZY2gC1

jZ1

ðlKEjÞ( )

: ð1:14Þ

The related Riemann surface can be realized as a twofold covering of thecomplex plane. To construct this covering, it is enough to take two copies of thecomplex plane with cuts corresponding to the spectral intervals

½E1;E2�;.; ½E2kK1;E2k�; ½E2gC1;NÞ;

and glue them along the cuts. After this, the cuts represent the transition linesfrom the upper sheet of the Riemann surface G to its lower sheet. The localparameters near the branch points PZ(0, Ej), P2G are tjZ

ffiffiffiffiffiffiffiffiffiffiffiffiffilKEj

p. At

‘infinity’ the local parameter is tNZ 1ffiffil

p . At the other points, we can take theprojection p(P)Zl as a local parameter. In Its & Matveev (1975a,b), we provedthat j(x, P) is a single-valued analytic function on G except for gC1 points. Theonly singularities of j(x, P) are g simple poles P (forming the non-special divisorDZP1CP2C/CPg) whose projections on the complex plane are the Dirichleteigenvalues: pðPjÞZljð0Þ, lying in the closures of spectral gaps. In addition, thefunction j(x, P) has an essential singularity of exponential nature at infinity

jðx;PÞZ eiffiffil

px ½1COð1=

ffiffiffil

pÞ�; P/N:

The two Bloch solutions j1,2(x, l) can now be considered as the projections ofj(x, P) to the upper and lower sheets of G.8

The x -dependent part of the product of two Bloch solutions, corresponding tothe finite-gap periodic potential, is a polynomial F(x, l) of l of order equal to thenumber of the non-degenerate gaps. The inverse statement, saying that theexistence of two solutions such that their product is a polynomial of l with x -periodic coefficients means that the related potential is periodic with g-gapswhere g is the order of the polynomial, is also true.
8Akhiezer in his seminal work (Akhiezer 1961) proved a similar uniqueness statement in a morespecial context, corresponding to a more restricted variety of pair finite-gap potentials. Later, Itsfound an explicit formula for j(x, P) in terms of theta functions using the fact that for given non-special divisor of the degree g the Riemann theta function, for which D represents its divisor ofzeroes always exists (see the formula (1.24) below). From this formula it is possible to find thecorresponding potential u(x), thus obtaining the solution to the inverse problem stated above. Thisstrategy has a far reaching extension for solving nonlinear integrable PDEs, especially emphasizedlater by the appearance of Krichever’s scheme for solving the KP equation. However, historically itwas not the first way of getting an explicit description of the finite-gap potentials.


V. B. Matveev846


In order to prove the first part of this statement (see Its & Matveev (1975a) forthe complete proof ), it is enough to study the solution f(x, t, l) of theSchrodinger equation with T-periodic g-gap potential u(xCt) fixed by theboundary conditions fð0; t; lÞZ0;fxð0; t; lÞZ1. There are g Dirichlet eigen-values lj(t) depending non-trivially on t. They are the simple roots of the entirefunction f(T, t, l). It was shown in Its & Matveev (1975a) that this satisfies thethird-order differential equation

LðtÞfZ 4lf;

where9

LðxÞZKv3x C2ðuðxÞvx CvxuðxÞÞ:

From the other side, it is well known (first mentioned by Hermite) that, for anytwo solutions f1,2(x,l) of the Schrodinger equation with any potential u(x), theirproduct y(x, l)Zf1f2 satisfies the same equation as above:

LðxÞyðx; lÞZ 4ly: ð1:15Þ

Thus, in the periodic case, the product of two Bloch solutions, being the only x -periodic solution of equation (1.15), is proportional to f(T, x, l), and hence to thepolynomial F(l),

FðlÞdYgkZ1

ðlK ljðxÞÞ:

This fact has a number of significant consequences. First, it leads immediately(Its & Matveev 1975a) to a very compact description of the ordinary differentialequations satisfied by the finite-gap potentials in terms of the recursion operatorL(x): all g-gap periodic potentials are the solutions of the nonlinear differentialequation

LðJ$LÞg1Z 0; J Z vK1x : ð1:16Þ

Next, it leads to the simplest (a few lines) proof of the fact that the Lamepotential gðgC1ÞPðxÞ has exactly g gaps (Its & Matveev1975b).

But the most important is that the same polynomial structure of the productof two Bloch solutions allows (see Its & Matveev (1975a,b) for details) toestablish the analytic properties of j(x, P) described above, and to reduce thecalculation of all the symmetric functions of lj(x) (and, hence, due to the traceformula (1.9), of all the finite-gap potentials) to the solution of the Jacobiinversion problem on a hyperelliptic surface (1.14). The solution of the Jacobiinversion problem allows (Its & Matveev 1975a) one to express the g-gappotentials and the solutions of the KdV equation corresponding to the finite-gap
9 It is worth mentioning that the same L operator plays many other fundamental roles in the theoryof the KdV equation (especially in its Hamiltonian aspects), providing a very compact descriptionof the conservation laws. It is also very important in the description of versal deformations of Hill’sequation obtained by Lazutkin & Pankratova (1975). Their work was later used by Witten (1988)in his further investigation of the coadjoint orbits of the Virasoro group.




initial data by means of Riemann g-dimensional theta functions (see the formula(1.23) below).

Let

dUj ZCj1l

gK1 CCj2lgK2.CCjgffiffiffiffiffiffiffiffiffiffi

QðlÞp

be the normalized Abelian differentials, associated to some canonical basis ofcycles aj,bj on G (see Its & Matveev (1975b, 1976) for details), and let B be therelated matrix of b-periods, that is

#ak

dUj Z dkj ; Bjkd#bk

dUj j Z 1;.g: ð1:17Þ

The Abel map of the divisor D and the vector of the Riemann constants, K,associated to G are defined by the formulae

UðDð0; 0ÞÞdXgjZ1

UðPjð0; 0ÞÞ; U jðPÞZðPNdUj ; P2G;

K j Z

jKPgkZ1

Bkj

2:

Let us consider lj(x, t), defined in §1c (and hence belonging to the closures ofthe spectral gaps), as the projection of the points Pjðx; tÞ2G coinciding with Pj

when xZ0, tZ0, i.e. pðPjðx; tÞÞZljðx; tÞ. Then it can be proved that the Abelmap of Dðx; tÞZ

PgjZ1 Pjðx; tÞ is a linear function of x and t

UðDðx; tÞÞZUð0; 0ÞCxpC tv; ð1:18Þ

pj Z 2iCj1; vj Z 8iðCj1C CCj2Þ; 2C ZX2gC1

kZ1

Ek :

Of course, the last equality should be understood as modulo periods of thedifferentials dUj.

The problem of reconstructing D(x, t) from D(0, 0) is known as the Jacobiinversion problem. Equation (1.18) shows that the KdV flow is described by thestraight line on the Jacobian of the curve G. There are different ways to deriveequation (1.18). For tZ0, it was derived in Its & Matveev (1975a,b) extendingAkhiezer arguments, i.e.makinguse of the single valuedness ofj(x, P) represented interms of Abelian integrals of the second and third kind. At the same time, Dubrovinderived it in another way, also for the case ts0. Indeed, Dubrovin was the first toobtain equation (1.18) for ts0. The main tool in his derivation was the followingsystem of autonomic differential equations for lj(x, t), nowadays known asDubrovin’s equations:

vxljðxÞZK2igðjÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQðljðxÞÞ

pQksjðljðxÞÞK lkðxÞÞ

hRj ;

QðlÞZY2gC1

mZ1

ðlKEmÞ; j Z 1;.; g; ð1:19Þ



V. B. Matveev848


whereg( j )ZG1.The evolutionof lj (x)with respect to theKdVflow is described inasimilar way by the following Dubrovin equation:10

vtljðx; tÞZK4Rj

Xksj

ljKC

!; 2C Z

X2gC1

kZ1

Ek : ð1:20Þ

The data ðg; ljð0; 0ÞÞZPj ; jZ1;.; g describe the non-special divisor DZP1C.CPg;Pj2G and the system (1.19) and (1.20) describes its evolution,Dð0; 0Þ/Dðx; tÞ, with respect to the space translations of the potential andto the KdV flow. The Abel map linearizes Dubrovin’s equations. Conversely,Dubrovin’s equations follow from equation (1.18): it is enough to differentiatethe r.h.s. of equation (1.18), and to use identities following from the Lagrangeinterpolation formula, in order to obtain the system (1.19) and (1.20).

All the symmetric functions of lj(x) might be found explicitly from the solutionof the Jacobi inversion problem. It follows from the trace formula (9) that thepotential u(x) (or more generally its evolution with respect to the KdV flowu(x,t)) up to a factor K2 and up to adding the sum of the spectral boundaries isthe simplest symmetric function of lj(x), their sum. To calculate this sum, andmore generally the power sums, Xg

kZ1

lmj ðx; tÞ;

the Riemann theta functions can be used. The g-dimensional theta functioncorresponding to the given g!g matrix B is defined by the formula

QðhÞdXk2Zg

expfpðBk; kÞC2pðh; kÞg; ðh; kÞdXgjZ1

hjkj :

Below, the matrix B is defined by equation (1.17) that guarantees it issymmetric and its imaginary part is positively defined. In this case, QðhÞ is anentire function of h2Cg.11 Let G be the variety with oriented boundary vG

10Historically, the fact that the existence of two solutions of the Schrodinger equation, whoseproduct is a polynomial of l, leads immediately (and with no other assumptions about theproperties of potential u(x)) to the trace formula (1.9), to the system (1.19) and to the Jacobiinversion problem to reconstruct the related potential, was first discovered by Jules Drach (1919);see also Belokolos et al. (1994) and the article by Brezhnev (2008). Drach, somehow, completelyignored the spectral aspects of the problem. He also did not obtain the final explicit theta-functional representation for the potential. It was not clear from his work that all smooth real-valued finite-gap periodic potentials are included in his construction. Probably, for this reason, hisremarkable work became known to the integrable community only around 1980 and, unfortunately,played no role commensurate with its value in the modern evolution of the theory. More specificstructure, corresponding to the existence of solutions of the Schrodinger equation, having the formj(x, k)ZP(x, k)ekx, lZk2, where P(x, k) is a polynomial in k-variable, corresponds to a degeneratecase of finite-gap potentials, known as Bargmann potentials. Bargmann studied only the case ofsmooth reflectionless potentials. In fact, the class of Bargmann potentials is pretty large. Itcontains, in particular, all the reflectionless potentials, singular periodic and multi-periodicpotentials, slowly decreasing and oscillating singular positon potentials (Matveev 2002) andrational singular potentials. For the Bargmann potentials, the product j(x, k)$j(x, Kk) is indeed apolynomial of l. Therefore, all the Bargmann potentials form a degenerate subfamily in the spaceof all finite-gap, almost periodic, potentials.11Of course the matrix B depends on the choice of canonical basis of cycles ak, bk.





obtained by cutting G along the cycles ak,bk. The calculation of the integralðvGlmd log QðUðPÞCxpC tvC lÞ; ldpðPÞ

leads to the following identities:

XgkZ1

lmj ðx; tÞZXgkZ1

#ak

lmdUkðlÞKresflmd log QðUðPÞCxpC tvC lÞgjN:

ð1:21ÞCalculation of the residues (Its & Matveev 1975b) in the right-hand side of

identity (1.21) (which takes in to account the local parameter in the vicinityof the infinity point on G is tN :Z1=

ffiffiffil

p), leads, for mZ1, 2, to the following

two identities:XgkZ1

ljðx; tÞZXgkZ1

#ldUk Cv2x log QðxpC tvC lÞ;

XgkZ1

l2j ðx; tÞZXgkZ1

#l2dUkK1

6v4x log QðxpC tvC lÞC 1

3vtvx log QðxpC tvC lÞ:

ð1:22Þ

Comparing the first of these identities with the first of the formulae (1.9), weobtain an explicit description of the g-gap periodic potentials and the relatedsolutions of the KdV equation:12

uðx; tÞZK2v2x log QðxpC tvC lÞC2CK2XgkZ1

#ak

ldUk; ð1:23Þ

ldKUðDð0; 0ÞÞCK ;

where the constant C and vectors p; v were defined in equation (1.18). Formula(1.23) for the finite-gap solutions of the KdV equation was first obtained in theworks (Its & Matveev 1975a,b).13

12 In our derivation of the identities (1.21) and (1.22), we followed closely the methodology of anexcellent review article (Zverovich 1971), where the solution of the classical Jacobi inversionproblem was clearly explained. Novikov told me later that when Dubrovin in Moscowindependently from us arrived to the Jacobi inversion problem to reconstruct the finite-gapperiodic potentials, they asked desperately the prominent algebraic geometrists how to solveefficiently the Jacobi inversion problem, and the typical answer was that there exists thebirational isomorphism between the gth symmetric power of the algebraic curve and its Jacobian,etc. The answer was certainly correct but far from enough to explain how to translate it to thelanguage of formulae. At that time the majority of the algebraic geometry community was livingtheir own life very far from physical applications, and many classical values had been completelyforgotten.13 Formula (1.23), being our invention, was first published in Dubrovin & Novikov (1974), with areference to our work (Its & Matveev 1975a), which was actually written before Dubrovin &Novikov (1974), but it was published only in 1975. May be, for this reason, in some articles (e.g.Airault et al. 1977, p. 137), it was attributed to Dubrovin & Novikov (1974).



V. B. Matveev850


Formula (1.23), with tZ0, reconstructs the finite-gap potential from theboundaries Ej of the continuous spectrum (i.e. the hyperelliptic curve G definedin (1.14)) and the divisor DZ

PgnZ1 Pj ; of poles of j. This divisor fixes the

eigenvalues of the Dirichlet problem at tZ0, ljð0; 0ÞZpðPjÞ, belonging to theclosures of non-degenerate gaps.

It is important to mention that the spectral approach was explaining how tochoose the parameters in formula (1.23) in order to get all real-valued non-singular x -periodic finite-gap potentials and the related solutions of the KdVequation; the branch points Ej should be distinct real numbers and theprojections lj(0, 0) of the points Pj must belong to the closures of the non-degenerate spectral gaps [E2j,E2jC1].

Formula (1.23), representing the finite-gap solutions of the KdV equation as asecond derivative of the Riemann theta function of the hyperelliptic Riemannsurface of the curve G, was very typical; the same kinds of formulae describe thefinite-gap solutions of the whole KdV hierarchy and of the KP hierarchies (shownlater, respectively, by Dubrovin and Krichever). In particular, the action of thehigher KdV flows on u(x, t) boils down to adding vector-valued functionsdepending linearly on the ‘higher’ time variables (with vector coefficients,depending only on G), in the argument of theta function in formula (1.23).

(e ) Theta-functional solution of the Schrodinger equation with finite-gappotential: the Alexander Its formula

The next important step was stimulated by the same work (Akhiezer 1961)containing the explicit formula for the j function in the case of a one-gap Lamepotential. It was natural to conjecture that an explicit solution of the Schrodingerequation with arbitrary genus g finite-gap potential should exist. This problem wassolved by Its at the beginning of 1975, at first for the periodic case.

Let U(P) be the normalized Abelian integral of the second kind, uniquelydefined (modulo b-periods) by the conditions

UðPÞZffiffiffil

pð1Coð1ÞÞ; P/N;P2G; #

ak

dUZ 0 k Z 1;.N :

The Abelian integral U(P) can be constructed explicitly

UðPÞZðPE2NC1

lN Cr1lNK1C.CrN

2ffiffiffiffiffiffiffiffiffiffiQðlÞ

p dl;

where Q(l) was defined in (1.14). The coefficients rk are determined from thesolution of the linear system arising from the above normalization condition.

Then, the following formula, found by Its, describes all solutions of theSchrodinger equation:

Kjxxðx;PÞCuðx; tÞjðx;PÞZ ljðx;PÞ; lZpðPÞ P2G;

with potential (1.23)

jðx; t;PÞZ eixUðPÞQðxpC tvCUðPÞC lÞ$QðtvC lÞQðxpC tvC lÞ$QðtvCUðPÞC lÞ : ð1:24Þ





Despite the fact that the Abelian integrals participating in this formula aremulti-valued functions on the curve G, the whole thing is a single-valued functionof P on G.14 Its projections on the complex plane form a fundamental system ofsolutions, j1,2(x, P) (except at the points PZ(0, Ej) to the Schrodinger equation,and their Wronskian Wdj1j

02Kj2j

01; is given by the formula

W ðt;PÞZ K2iffiffiffiffiffiffiffiffiffiffiQðlÞ

pQðlK ljð0; tÞÞ

:

For the proof see Its & Matveev (1976). When P/N,

jðx; t;PÞZ eixffiffil

pð1Coð1ÞÞ;

and when tZ0 its poles coincide with Pj(0,0). If ts0 the poles of j coincide withPj(0, t).

15

(f ) From the periodic to the almost-periodic case

For a generic choice of parameters Ej defining G, the right-hand side of formula(1.23) is obviously not periodic, but almost-periodic and, in general, is a complex-valued function of real the variables x and t. Therefore, it was natural to supposethat the right-hand side of formula (1.23) remained the solution of the KdVequation, but the periodic spectral theory could not be applied to prove it.

Soon we obtained two different rigorous proofs of the fact that the formula forthe solutions of the KdV equation, which we obtained for the periodic case, canbe extended to the case when G is an arbitrary non-singular hyperelliptic curve ofgenus g, and D is an arbitrary non-special divisor on G, deg(D)Zg. We provedthat formula (1.24) for the solution of the Schrodinger equation is valid, not onlyfor the periodic case, with smooth real potential, but also for the same genericdata G.

It was, probably, the first case when the large class of almost-periodic spectralproblems was explicitly solved.16

The first proof, which I proposed, used a comparison of ‘linear’ and ‘nonlinear’trace formulae. Amazingly, in this proof the Schrodinger equation was
14 Indeed, the factors gained by the exponential and theta-functional parts of the right-hand side of(1.24) when passing along the basic cycles cancel each other. Therefore, to determine the right-hand side of (1.24) it is enough to assume that the contours of integration in none the Abelianintegrals involved cross the basic cycles.15 Formula (1.24) for the j appeared to be even more generic than the formula (1.23) for thepotential. With small variations, it appears in representations of the matrix elements of thej-function associated to more complicated models (Belokolos et al. 1994). It is also worthindicating the following important difference of formulae of the type (1.24) when compared withthe traditional theta-functional representations of rational functions on algebraic curves (Mumford1983). The latter assume a prior knowledge of both poles and zeroes of the function, while in (1.24)only the poles are explicitly involved. The knowledge of zeros is equivalent to knowledge of anexplicit solution of the Jacobi inversion problem. This explains why the traditional formulae arenot of great help in finite-gap integration.16Novikov told me at that time that the same formulae in the almost periodic case might be provedrigorously using the Liouville algorithm. However, this kind of proof was published eventually onlyin 1979 by Gelfand & Dickey (1979).


V. B. Matveev852


completely eliminated. The heart of the proof is to establish two relations

vx

XgkZ1

l2j ðx; tÞZ1

12uxxxK

1

6ut;

which can be called a ‘linear trace formula’ (following immediately from thecomparison of the right-hand side of the second identity (1.22) with (1.23)), andthe identity which I called a ‘nonlinear trace formula’

vx

XgkZ1

l2j ðx; tÞZ1

2uuxK

1

4ut:

The comparison of the right-hand sides of these two identities proves that thefunction u(x,t), defined by the formula

u ZK2XgkZ1

ljðx; tÞCX2gC1

kZ1

Ej ;

represents a solution of the KdV equation.In this formula, the l(x, t) are defined as the projections on the complex plane

of the solutions Pj(x, t) of the Jacobi problem (1.18). The initial divisor D(0, 0)was supposed to be any non-special divisor of degree g on G, and G was supposedto be any non-special hyperelliptic curve of the form (1.14). Quite obviously, anexplicit formula for u(x, t) might be proved as before. In order to prove thenonlinear trace formula it is enough to write the second Dubrovin equation(which follows from equation (1.18)) as ðljðx; tÞÞtZ2ðljÞxðuðx; tÞC2ljÞ and thenperform the summation over j from 1 to g. The details can be found in Its &Matveev (1976).

The second proof of the fact that the same formulae are valid for a generichyperelliptic curve and non-special divisor on it amounts to checking that theassociated formula (1.24) for the j-function is valid also in a generic situation,that is, for the potential u(x, t) constructed by the same formula, starting fromgeneric data (G, D). This produces another kind of nonlinear trace formula(Matveev 1975), allowing a check that u(x, t) is again the solution of the KdVequation (Its & Matveev 1976). All these results were first reported in my talk atthe Petrovsky seminar in Moscow in April 1975, and published in the last issueof 1975 of the Russian journal Uspekhi Math. Nauk (Matveev 1975) known inEnglish translation as Russian Math. Surveys. Unfortunately, it was the lastissue of the journal for which part of the content reproducing thecommunications of Petrovsky seminar was not included in the Englishtranslation. Its content was reproduced later in a review article (Dubrovinet al. 1976), which we wrote with Dubrovin and Novikov, and in a more detailedversion in Its & Matveev (1976).

In the latter article, see also Matveev (1976), it was also shown how to getexplicit formulae for multi-soliton solutions from formula (1.23), by contractingthe branch points of the spectral curve in such a way thatE2jK1;E2j/aj ; a1!a2;.!ag!0. In this limit the diagonal elements of thematrix iB, where B is the matrix of b-periods, tend to KN. The limit of the





non-diagonal elements of B is given by the formula

lim Bjk Zi

plog

nj Cnk

njK nk

��; nk Z

ffiffiffiffiffiffiffiffiffiKak

p; jsk j; k Z 1;.; g;

in which it is immediate to recognize the asymptotic phase shifts characterizingthe large time asymptotics of the multi-soliton solutions of the KdV equation.Owing to the presence of the elements of the matrix B, both in the definition ofthe theta function and in the components of the vector l, in such a limit the thetafunction in formula (1.23) transforms into a finite sum where the summation istaken over the finite number of g-dimensional vectors k; kjZ0; 1. The form of themulti-soliton solution obtained by this passage to the limit reproduces the well-known Hirota representation for the multi-soliton solutions. The related initialdata for this solution is a reflectionless potential, having discrete negativeeigenvalues at the points aj. This gives a precise meaning to the claim that finite-gap potentials represent the multi-periodic generalization of the reflectionlesspotentials, and that the finite-gap solutions of the KdV equation are the naturalalmost periodic extensions of multi-soliton solutions.

The partial degeneration in the formulae of the finite-gap integration might bedescribed quite differently. It is possible to perform the degeneration in the genericformulae, or to use the Darboux dressings (Matveev 1979a) or to use an axiomaticdescription of the degenerate Baker–Akhiezer function (Krichever 1975). Thesethree approaches lead to different descriptions of the same solutions.

Summarizing, the explicit construction of the j-function from the algebro-geometric data (G, D), and its application to the integration of the KdVequation, free from the restriction on the spectral data, imposed by the conditionof smoothness, reality and periodicity of the potential, was completely workedout in the beginning of 1975.

Getting an explicit solution for the Schrodinger equation with finite-gappotential was an important by-product of this activity, having the same value asthe solution of the periodic KdV initial problem. For the solid-state physicscommunity, this provided large classes of periodic and almost periodic potentials,for which the related Schrodinger equation was explicitly solvable. This class ismuch better adapted to approximating real physical situations than the modelsof Kronig–Penny type.

The Its formula (1.24) later found many different applications to finite-dimensional dynamical systems (geodesics on an N-dimensional ellipsoid, couplednonlinear oscillators, etc.) in the works by Veselov (1980) and Knorrer (1982)and some other researchers.

The Hamiltonian aspects of the finite-gap integration were developed in aseries of beautiful papers by Novikov, Dubrobin, Bogoyavlenskij and also byGelfand and Dickey. See, for instance, the recollection of review articles fromRussian Math. Surveys (Wilson 1981).

(g ) Passage to 2C1 integral systems. KP and Krichever’s work

The next very important development of the method of finite-gap integrationwas the passage to the integration of 2C1 KP-like systems, realized by Kricheverin 1976. He extended our method of using the j-function for integrating the KdV



V. B. Matveev854


equation in two senses. First, he observed in Krichever (1975) that the non-stationary asymptotic condition

jðx; t;PÞZ expðikxK4ik3tÞ½1Coð1Þ�; k/N;

where PðkÞ2G and kK1 is a local parameter at infinity, specifying the solutionjðx; t;PÞ of the Lax system

LjZ lðPÞj; vtjZAj;

has an important advantage. Namely, for this asymptotic condition the divisor ofpoles of j becomes t-independent. Then it is easier to prove that j(x, t)(reconstructed from the same kind of data as in our approach) solves the Laxsystem with coefficients expressed via the first terms of its asymptotic expansionat infinity. We used the stationary normalization (see the text above) ofj(x, t, P), which is less comfortable for working with evolution equations. Ourway of checking that j(x, P), constructed via the Its formula, satisfies theSchrodinger equation with potential corresponding to generic algebro-geometricdata was considerably longer (Its & Matveev 1976), although the main idea wasthe same.

The next remarkable observation of Krichever was that replacing the firstequation in the Lax system by the evolution equation

LjZ vyj;

one can solve it via the same kind of formula as in the previous case, replacing thehyperelliptic curve in the previous construction by any non-singular algebraiccurve G of genus g. The coefficient u(x, t, y) of the related Lax system is thesolution of the Kadomtcev–Petviashvily equation and, as in the KdV case, thissolution can be obtained easily from the second term of the asymptotics of j atthe marked point of the Riemann surface replacing the infinity point of theKdV case. The data, determining j(x, t, y, P), are the non-special pole divisorD (degDZg), the marked point P0, and the asymptotic condition at themarked point

jðx; y; t;PÞZ expðkxCk2yCk3tÞ½1Cxðx; y; tÞkK1 COðkK2Þ�; k/N; ð1:25Þ

where KK1 is a local parameter at the vicinity of the marked point P0.Quite similarly to the Schrodinger–KdV case, j(x, y, t, P) can be reconstructedexplicitly from the algebro-geometrical data (G, D, P0) in terms of thetafunctions and normalized Abelian integrals wj ; jZ1; 2; 3 of the second kind,with simple poles at the point P0, and principal parts equal to k, k2 and k3,respectively:

jðx; y; t;PÞZ exw1ðPÞCyw2ðPÞCtw2ðPÞ$QðxpC tgCyrCUðPÞCdÞ$QðdÞQðxpC tgCyrCdÞ$QðUðPÞCdÞ ; ð1:26Þ

where d is chosen in such a way that the poles of j coincide with the divisor D,and p, g, r are the vectors of b-periods of the Abelian integrals wjðPÞ divided bythe factor 2ip.





Krichever noticed that in order to check that j(x, t, y, P) is a common solutionof the system

LjZ vyj; AjZ vtj;

LZ v2x Cuðx; y; tÞ; AZ v3x C3

2uðx; y; tÞvx Cvðx; y; tÞ;

it is enough to check that j(x, t, y, P) solves it asymptotically,17 when P/P0,which can be achieved by taking uðx; y; tÞZ2vxxðx; y; tÞ:

Compatibility of this system implies that j(x, y, t) is solution to the KP equation

3uyy Z vxð4utK6uux Cu xxxÞ: ð1:27Þ

Combining these arguments, with an explicit representation for j, Kricheverderived the following formula for the solutions of equation (1.27):

uðx; y; tÞZ 2v2x log QðxpCygC trCdÞCC : ð1:28Þ

When w2(P) is a meromorphic function (which automatically means that theunderlying curve is hyperelliptic and the marked point P0 coincides with one of itsbranch points), u(x, y, t) loses its dependence on y and transforms into the solution ofthe KdV equation (modulo obvious rescaling with respect to the definition (4) of theKdV equation) as discussed above. Quite similarly, if w3(P) is a meromorphicfunction onG, whichmeans thatG is a trigonal curve, andP0 is one of itsWeierstrasspoints, the solution (1.28) becomes t-independent and reduces to the solution of theBoussinesq equation.

Of course, this ‘non-spectral’ approach produces in general complex-valuedand singular solutions. Isolating the smooth real-valued solutions neededseparate non-trivial work, which was mainly done by Dubrovin & Natanzon(1989). The same scheme was applied by Krichever to integrate the whole KP (orZakharov–Shabat) scalar hierarchy of equations obtained as compatibilityconditions of the linear evolution equations, with higher order differentialoperators (of mutually prime order) in the x variable on the right-hand side. Theonly modification needed was to replace k 2 and k 3 in the asymptotic condition(1.25) by arbitrary polynomials of k.

Owing to knowledge of the Baker–Akhiezer functions, it is possible to obtainlarger families of solutions to the KP equation, also expressed by means of theRiemann theta functions, by applying the dressing formulae (Matveev 1979a;Matveev & Salle 1991).18 Namely, for any m densities rjðPÞ jZ1;.m, definedon the algebraic curve G, and given solution (1.28), u(x, y, t) of the KP equation
17 In fact, the same strategy was proposed, but not finalized, for linear ordinary differentialequations of any order, by Baker in his article (Baker 1928), which became known later. Followingthe proposal of Novikov, nowadays expressions similar to (1.24) and (1.26) and their vector ormatrix valued generalizations are called Baker–Akhiezer functions.18 I proposed to call them Darboux dressing formulae, although the original Darboux results(Darboux 1882a) were valid only for dressings of the Sturm–Liouville equation. The extension ofhis approach to the hierarchies of linear and nonlinear PDEs and their lattice and non-Abelianversions was proposed in Matveev (1979a,b) and Matveev & Salle (1991), where numerousapplications to the theory of solitons can be found.


V. B. Matveev856


(1.28), the formula

vðx; y; tÞZ uðx; y; tÞC2v2x log W ðf1;.; fmÞ; ð1:29Þ

fj Z

ðGrjðPÞjðx; y; t;PÞdP ð1:30Þ

(where W means the Wronskian determinant and j(x, y, t, P) is a Baker–Akhiezer function associated with u(x, y, t)) gives a new solution of the same KPequation depending on m functional parameters rj(P). The obtained family ofsolutions is very large and is still not completely studied. Its connection with theso-called higher rank solutions, introduced by Krichever & Novikov (1980) andKrichever & Novikov (2003) for the discrete systems, is not completelyunderstood, to my knowledge, until now.

Using equation (1.29), we can assume that uZu(x, t) is a given solution of theKdV equation, and the related Baker–Akhiezer function depends on yexponentially. In this way, we arrive to generate the solutions of the KPequation starting from a given solution to the KdV equation. A similar remarkapplies also to the generation of the solutions of the KP equation from thesolutions of the Boussinesq equation, given the assumption that uZu(x, y) and

the dependence of j(x, y, t, P) on t is purely exponential, i.e. rZ0.19 Furtherresults concerning the periodic closures of the chain of Darboux transformationscan be found in Takasaki (2003).

It is interesting to mention that for the cylindrical KP equation, also known asthe Johnson equation

3uyy

t2Z vx 4utK6uux Cuxxx C

u

2t

� �;

there exists a family of theta-functional solutions that are still periodic or almostperiodic functions of x and t, but no longer periodic or almost periodic withrespect to y (Lipowskij et al. 1986; Matveev & Salle 1991), which completelychanges the qualitative properties of the solutions with respect to the KP case.The interested reader can find many more details and graphic images of somemost interesting solutions describing complicated wave interaction with familiesof crossed parabolic fronts in a recent article (Klein et al. in press), where thegauge equivalence of the related Lax pairs is also explained.

The natural question concerning whether one can find an explicit solution to theJohnson equation that tends to some periodic or almost periodic solution to theKdVequation, to my knowledge, remains open.

19 In the one-dimensional Schrodinger case one of the ways to isolate the finite-gap solutions(discovered relatively recently by Shabat & Veselov (1993) and Yamilov) is to look for periodicclosures of the chains of Darboux transformations. A similar theory for the case when the Sturm–Liouville equation is replaced by a non-stationary Schrodinger equation or by a heat equation thatdoes not exist, to my knowledge, although the formalism of the Darboux transformation methodwas generalized long ago to much larger classes of linear and nonlinear PDEs and their non-Abelianand difference versions (Matveev 1979a,b; Matveev & Salle 1991). For possible generalizations ofShabat & Veselov (1993) to the PDE case, see Novikov & Dynnikov (1997), where the simplestcase of the period 2 closure of the Darboux-dressing chain for the heat equation was discussed.





(h ) Dirac and Baker in Cambridge

Explicit formulae for Baker–Akhiezer functions, and explicit statementssimilar to formulae (1.23)–(1.28), were not known before, although concerningthe solutions of the linear problems in the strategic point of view, saying that itis possible to reconstruct certain kinds of single-valued functions from thealgebro-geometric data, was mentioned by Baker (1928) as a comment toremarkable works by Burchnall & Chaundy (1923, 1928).20 Baker noticed thatthese functions, which can be considered as higher genus analogues ofexponentials, can be used for solving linear differential equations with coefficientsthat can be calculated in terms of coefficients of the asymptotic expansion ofthese functions in the vicinity of some marked point on the algebraic curve.Somehow, as in the case of Jules Drach, his work was not widely known and incontrast to Akhiezer’s work was rediscovered too late to play some role in thecreation of the finite-gap integration method, as happened also with the worksof Burchnall & Chaundy.21

It is not widely known that around 1926–1927 Dirac was regularly meetingBaker to participate in his scientific tea-parties at Cambridge. Dirac claims,in his inspiring article (Dirac 1977), that projective geometry, a main topicat the Baker tea-parties, was very important for his vision concerning therole of beauty in the mathematical description of reality. He also explainsthat the projective geometry viewpoint was often hidden behind his ownquantum mechanical discoveries owing to the necessity of adapting thepresentation to the mathematical background of the physical community ofthat time. The first scientific communication made by Dirac in Cambridgeconcerned the proofs of some theorems in Euclidean geometry by projectivemethods. But soon Dirac left Baker’s seminar. He also never attended hislectures ‘since Baker was doing geometry’ and Dirac was fully involved in thedevelopment of quantum mechanics. The period during which Baker wrotehis article (Baker 1928) was certainly close to the time when Dirac, at firstsceptical about Schodinger’s formulation of quantum mechanics, becameinterested in it. He certainly ignored Baker’s activity on differentialequations and Abelian functions; otherwise the story of finite-gap integrationmight have been very different.

20As a matter of fact, in his monograph Baker (1897, 1907) introduced certain multi-valued objectson a Riemann surface, which he called ‘factorial functions’, and which are characterized bymultiplicative jump conditions across a and b cycles. Moreover, Baker showed that the factorialfunctions can be expressed in terms of Riemann theta functions using essentially the samearguments as the ones used to derive explicit formulae of the type (1.24) and (1.26). Indeed, thetheta functional part of the right-hand side of Its formula (1.24) is a factorial function.21 The tendency, appearing in certain works, to call the Baker–Akhiezer functions just Bakerfunctions should be understood as a late recompense for his pioneering ideas, although I do notthink that elimination of Akhiezer’s name is a good idea. Akhiezer’s work played a crucial role inthe creation of the modern finite-gap integration method. In fact, Akhiezer first found theconnection between inverse spectral problems and the algebraic geometry of Riemann surfaces,relevant to finite-gap Schrodinger operators. He was also the first (Akhiezer 1960) to producesimilar work in the discrete case in connection with the construction of polynomials orthogonal ona system of intervals, relevant to the integration of Toda-like lattice systems.



V. B. Matveev858


(i ) Matrix differential operators, zero-curvature representationsand finite-gap integration

Parallel developments were coming mainly from several directions. First, itwas connected with the extension of the finite-gap integration method to thecases connected with Lax pairs containing matrix differential operators or, moregenerally, zero-curvature representations with rational or elliptic dependence onthe spectral parameter. This extension of theory was relevant for constructingthe finite-gap solutions for physically important models like NLS, sine-Gordonequation, Landau–Lifschitz equation (see Belokolos et al. (1994) for detailedexposition and further references), vacuum axially symmetric Einstein equationsof general relativity (Korotkin 1988; Korotkin & Matveev 1988, 1990, 1999,2000; Frauendiener & Klein 2001, 2004) corresponding to the special dependenceof the moduli of the underlying spectral curve on space and time variables, andfor many other models describing quite interesting physical situations.

For developing the general algebro-geometrical point of view on the modelsconnected with matrix-differential Lax pairs, the analysis of the NLS model (firstsolved in the ‘finite-gap’ way by Its (1976) in the ‘repulsive’ case, and by Its &Kotlyarov (1976) in the attractive case) and the sine-Gordon model (Kozel &Kotlyarov 1976) was certainly very important.22 Analysis of the NLS model andthe sine-Gordon model gave the precise idea about the difference betweenanalytical properties of the vector-valued Baker–Akhiezer functions, and theirscalar counterparts, by reason of introducing the very first examples of the multi-points Baker–Akhiezer functions. This together with the important work byDubrovin (1976) on general matrix finite-gap differential operators built a basefor the development of the matrix version of Krichever’s scheme. In fact, the firstexplanation of how it is possible to integrate the NLS equation, the modifiedKdV equation and the sine-Gordon model, in the spirit of Krichever’s scheme,was presented in my Polish lectures (Matveev 1976), based in part on someunpublished works by Its.

Soon after, Krichever (1976b) generalized his approach to generic hierarchiesof the nonlinear Zakharov–Shabat type equations connected with matrixdifferential operators. This ‘generic integration’ scheme, though very impressive,did not include many of the physically interesting models such as the sine-Gordon equation, the modified NLS equation (Its & Matveev 1984), the Kaup–Boussinesq equation (Matveev & Yavor 1979), the Landau–Lifshitz equation(Bobenko 1985), the Ernst–Einstein equation of general relativity (Korotkin &Matveev 1990; Korotkin 1996) and many other models where very subtle worktaking into account the necessary reductions on the parameters has been done.

(j ) Discrete Toda-like models, polynomials orthogonal on a systemof intervals and finite-gap integration

The background to solving the discrete periodic problems associated with theToda-like or discrete KdV-like integrable systems was, in principle, provided bythe article by Akhiezer (1960), which, for some mysterious reasons, is cited in theliterature much less often than Akhiezer (1961). In Akhiezer (1960), theconstruction of the system of polynomials orthogonal on the finite number of
22 These works were written in the middle of 1975 but related results were published only in 1976.




intervals was reduced to the solution of the Jacobi inversion problem on thehyperelliptic curve, and the analytical properties of the related solution of thediscrete Schrodinger equation were also established, although explicit theta-functional formulae for those polynomials were unknown at that time.23 Amongthe old works written before the creation of the finite-gap integration method,one should mention also the result by Naiman (1962), see also Glazman (1963),describing the spectrum of generalized complex-valued periodic Jacobi matricesand the discrete analogue of the theorem by Burchnell–Chaundy, concerning thedescription of commutative rings of periodic difference operators. One of theobvious conclusions suggested by these results was the fact that discrete periodicdifference operators have always a finite-gap spectrum formed in general by afinite number of arcs in the complex plane. In the self-adjoint situation, thesearcs become finite closed intervals of the real line. Therefore, for discrete systemsthe finite-gap structure of the spectrum corresponds to the generic periodic case.

During the period 1974–1976, many important aspects of the finite-gapintegration method for the discrete Toda-like models were worked out (Kac &van Moerbeke 1975; Date & Tanaka 1976; Dubrovin et al. 1976; Flaschka &McLaughlin 1976). It was clear from the spectral theory of the Jacobi matricesthat for the discrete models the finite-gap solutions contain all solutions periodicwith respect to the discrete (lattice) space variable. Again, generic solutions arealmost periodic but the isolation of the purely periodic solutions from the genericcase is easy, and efficient, contrary to what we have in the continuous models.For instance, in the Toda lattice case, the condition of n-periodicity suggested bythe spectral theory of difference Shrodinger operator is that the relatedhyperelliptic curve should be defined by the equation

w2 ZP2ðzÞCconst:;

where P is a polynomial in z. Explicit finite-gap solutions of the Toda lattice can beobtained along the same lines as in the KdV case; after the works by Kac & vanMoerbeke (1975), Dubrovin et al. (1976) andDate&Tanaka (1976), it was finalizedby Krichever (1978). It is also necessary to mention here the important work byFlaschka & McLaughlin (1976) where the action-angle variables for the periodicToda lattice were constructed. Many additional important results, concerning thefinite-gap integration for discrete systems can be found in Mumford & vanMoerbeke (1979), Krichever (1981) and Krichever & Novikov (2003).

2. Some further developments of finite-gap integration after 1976

(a ) Reductions to the lower genera, elliptic solitons, effectivizationand Schottky problem

From the very beginning of the appearance of the theta-functional formulae, suchas (1.23) for the solutions of the KdV equation, there had been a discussion abouttheir efficiency. I remember well that after the appearance of formulae (Its &Matveev 1975a) for the solution of the KdV equation, Novikov’s comment was
23 For more complete references on the Akhiezer and Tomchuk works, and the related advancedresults in this direction, relevant in particular to the theory of matrix models, see Chen & Its(2008).


V. B. Matveev860


that it was of extreme level of complexity and not efficient for calculations.Nowadays, we understand that formulae of the type (Its & Matveev 1975a,b;Krichever 1976a) are perfectly adapted for performing numerical calculationsand for producing the plots of solutions (Bobenko & Bordag 1987, 1989). But, atthat time the authority of Novikov pushed people to look for different answers tothis challenge.

First, the idea pursued by many people was to look at special initial data, suchas the 2-gap or g -gap Lame potential. In particular, Airault et al. (1977) hadshown that the common action of the KdV and higher KdV flows on the Lamepotentials produced the whole variety of g-gap solutions elliptic in the x variable.By contrast, the t dependence of the same solutions was rather complicated andthe evolution of the poles xj(t) was isomorphic to the trajectories of the classicalmulti-particle higher Calogero–Moser-like systems. So, despite the appearance ofthe very interesting connections with finite-dimensional integrable systems therewas no gain in efficiency of the solutions. Similar work in the KP case was doneby Krichever (1980). In his work, the link between the elliptic in x solutions ofthe KP equation and the trajectories of the elliptic Calogero–Moser-like systemsrealized by y-dependence of poles xj(y) was discovered.

The solutions of the KdV equation elliptic in the t variable and similarquestions for other nonlinear integrable equations, including NLS, the Todalattice and the sine-Gordon equation, were obtained by Smirnov (1994). A newapproach to isolating the elliptic finite-gap solutions, based on extensive use ofPicard’s theorem, was recently proposed by Gesztezy and Weikard (Gesztesy &Holden 2003).

The next idea, revealing many beautiful effects, was to choose as spectral curvescompact Riemann surfaces possessing the non-trivial group of birationalisomorphisms or having the appropriate covering structure. In many cases, itwas leading to a decomposition of the Riemann theta function of genus g to a sumof a finite number of terms representing products of a finite number of one-dimensional theta functions, keeping the linear dependence of their argumentswith respect to space and time variables. In such situations, it is often possible(Babich et al. 1983, 1985) to calculate the matrix of the B-periods explicitly,taking into account only symmetry arguments. For some recent, interesting,applications to quasi-periodic vortex structures in two-dimensional flows ofincompressible inviscid fluids, see the recent article by Babich & Bordag (2005).24

24 In some more exceptional cases, like the famous genus 3 Klein curve, the related Jacobian is notonly isogenous but also isomorphic to a product of three elliptic curves. Jean Pierre Serre in hisunpublished letter to M. V. Babich of 22 July 1985, commenting on our article (Babich et al. 1983)pointed out that until now it is not known if a similar phenomenon takes place for infinitely manyvalues of genera, and gave some other examples where the same effect occurs (the Briggs curve ofgenus 4, and some other curves of genus 26 and 43). Generic Riemann surfaces have no conformalautomorphisms for genus gR3. Any finite group can be realized as a group of conformalautomorphisms of some Riemann surface. For a given genus g, it was shown by Hurwitz that theorder of this group does not exceed 84 (gK 1). There is an infinite number of genera for which thismaximal order is realized. It is reasonable to call the related curves as Hurwitz curves. The firstinstances where Hurwitz curves do appear are gZ3 (Klein curve) and gZ7 (Fricke curve).Actually, it is known that for any given number n it is possible to find the genus g for which thereexist n conformally inequivalent Hurwitz curves. I believe such beautiful objects as Hurwitz curvesshould play a special role in the theory of integrable systems and this still has to be understood.





Another approach to the reductions of the special finite-gap solutions to one-dimensional theta functions exploring the Weierstrass programme of the reduc-tions of the Abelian integrals to elliptic integrals was proposed by Belokolos andEnol’skii (see Belokolos et al. (1986) for the related results and references).

Of course, solutions obtained in the cases where there exists a non-trivialgroup of conformal automorphisms of the spectral curve are much simpler fornumerical calculations with respect to the generic situation. The firstvisualization of the special genus 3 finite-gap solutions of the sine-Gordonequation using the aforementioned symmetry reduction approach was realized inthe article by Babich & Bordag (1985), using the results of Babich et al. (1985).I think at that time it was the very first work where plots of finite-gap solutionsof genus greater than 2 were obtained.

The interested reader can consult, for instance, the book by Belokolos et al.(1994) for further examples and references concerning applications of thesymmetry reduction approach. Among other things, this activity explained howto obtain the Lame equation as a reduction of the general finite-gap formulae.Some more complicated reductions of the generic finite-gap potentials (1.23)representing the four-parametric generalization of the Lame potentials werefound by Treibich (1989) and Verdier. They called the related solutions of theKdV equations elliptic solitons (in my opinion, a not very much justified name,which is, however, frequently used). Later, it was realized that the same equationwas first found and solved by Darboux (1982b, 1894) using an absolutely differentapproach. For further generalizations and more references see Smirnov (2002).More recently, the same potentials were rediscovered within the framework of aSUSY approach in works by Khare & Sukhatme (e.g. Khare & Sukhatme (2005)and the references therein). Additional material concerning this topic can befound in Belokolos et al. (1986, 1994), Gesztesy & Holden (2003) and Matveev &Smirnov (1990, 1993, 2006). In the last article, among other results (including theproof of the Baxter et al. (1988) normalization conjecture, see also Baxter(2002)), it was shown that the Riemann theta functions corresponding to (non-hyperelliptic) Ferma curves, parameterizing the Boltzmann weights, correspond-ing to the solvable chiral Potts model of statistical physics (the first quantummodel with spectral parameter ‘living’ on the curves of genus greater than 1), canbe split into a finite sum of finite products of elliptic theta-functions.

Remarkably enough, the discussion of the efficiency of the general theta-functional formulae also had a very important consequence for algebraicgeometry. This emerged in the course of pushing the original ‘effectivizationprogramme’ for finite-gap integration, proposed by Dubrovin and Novikov, whichis presented, for instance, in the review article by Dubrovin (1981). The idea therewas to consider the finite-gap-like formula for the solution, taking, instead of thematrix ofB-periods, any symmetric matrix with positively defined imaginary part,not reducible to a quasi-diagonal block structure by the Siegel transformations,and then directly substituting the so-obtained anzatz into the relevant nonlinearequation, and then solving the appearing dispersion equations for the vectorcoefficients in front of x, y and t in the arguments of the theta functions.

This programme fails already for genus 3 for 1C1 systems due to the existenceof the non-hyperelliptic curves in genus 3 and the absence of efficient criteria todistinguish the hyperelliptic 3!3 matrices of b-periods from those associatedwith non-hyperelliptic curves. For 2C1 KP-like systems where (as we know from



V. B. Matveev862


Krichever) all the finite genus curves are admissible, the same programme worksin the genus 3 case also, but it fails again starting from genus 4, owing to theappearance of new restrictions on the matrix of b-periods known as the Schottkyrelations. There are no efficient tools to determine if the given B-matrix(satisfying of course the necessary conditions of being symmetric with positivelydefined imaginary part and irreducible to the block form by the Siegeltransformations) is the matrix of b-periods of some compact Riemann surface.Indeed, this question constitutes the classical Schottky problem. This inspiredNovikov to formulate the following conjecture: ‘The given matrix B (satisfyingthe aforementioned necessary conditions), is a matrix of B-periods of an algebraiccurve if and only if the right-hand side of (1.28) is the solution of the KPequation.’ This conjecture was proved by Shiota (1986), who thus obtained whatnowadays is acknowledged as a solution of the Schottky problem. For a differentproof of Novikov’s conjecture see Arbarello & de Concini (1987).

Shiota’s proof of Novikov’s conjecture revealed a new and extremely excitingside of algebro-geometric integration. It showed that soliton theory not onlymakes use of algebraic geometry but can actually contribute to it. It showed thatintegrable PDEs, such as the KP equation, provide an adequate tool for studyingseveral highly non-trivial questions in the theory of algebraic curves.

This direction has been pursued further recently in a series of papers byBuchstaber et al. (1997), see for example their review article, and has led inparticular to a revival of the beautiful classical theory of multi-dimensionalKleinian z and P-functions.

With all its theoretical importance, the solution of the Schottky problemoffered by Shiota’s proof of Novikov’s conjecture does not provide an efficientalgorithm for selecting matrices of b-periods of algebraic curves out of genericsymmetric irreducible matrices with positive imaginary parts. Indeed, one canmake sure that a given theta-functional anzatz satisfies the KP equation only upto some numerical accuracy. It is efficient only to check whether a given matrix isnot a matrix of the b-periods of some Riemann surface. In other words, theefficient solution of the Schottky problem is still missing and it is not clearwhether it exists at all. In some special cases, however, it is possible to find thematrix of the b-periods explicitly (see Babich et al. (1983, 1985), Belokolos et al.(1994) and the references given there).

By contrast, the computation of the finite-gap solutions became much moreefficient than had been expected initially. In this respect, the positive evolutionwas approximately the following.

First, Bobenko tried to use the Poincare ideas to compute theta functions andAbelian integrals via general Poincare theta series. The result was an article(Bobenko 1983), where a new representation of the same finite-gap solutions ofthe KP and KdV equations in terms of the Poincare theta-series was obtained,but it was not helpful for performing numerical calculations.

Next, I remembered that Burnside (1890, 1892) used special classes ofautomorphic functions to solve some concrete problems of hydrodynamics (heconsidered the two-dimensional flows of ideal liquid in the presence of Ncylindrical obstacles). The related compact Riemann surfaces were parameter-ized by classical Schottky groups. The elements of the matrix of b-periods andthe normalized Abelian integrals were represented there in the form of aconvergent Poincare theta series of dimension 2. At that moment, I had only a





copy of the relevant chapter of the book (Baker 1897) that reproduced Burnside’sresults,25 which I gave to Bobenko, suggesting he look at these works to see ifthey could give something better than his first ‘automorphic attempt’. It wasreally the case. Soon after, Bobenko proved, using these works and some otherresults, that it is possible to parameterize all real non-singular solutions of theKP-II equation by the classical Schottky groups.26 The related numericalalgorithms were developed in Bobenko & Bordag (1987, 1989) and Belokoloset al. (1994). In particular, the plots of the genus 4 solutions of the KP-IIequation using the classical Schottky group parametrization were first obtainedin Bobenko & Bordag (1989). From that moment, it became clear that there is noproblem in calculating the solutions of the KP equation of any genus and toproducing related plots of the solutions.

Later it was understood that the original spectral parametrization for theKdV-like integrable equations works perfectly well for doing the numerics.Finally, the initial scepticism concerning the finite-gap formulae as a tool fordoing numerics completely disappeared.

It also became clear later that the automorphic approach is not actually thebest and in many cases direct spectral parametrization is better adapted fordoing numerics and producing plots. Nowadays, even in much morecomplicated situations where the moduli of the spectral curve depend onspace and time variables (e.g. this is the case of vacuum axially symmetricstationary solutions of the Einstein equations discussed in §2b), the finite-gapformulae are still good for numerics, as was shown recently by Frauendiener &Klein (2001, 2004, 2006) and Klein & Richter (2005). The related solutionplots can be obtained on a standard PC using the standard MATLAB program.More generally, for the whole class of algebro-geometrical solutions of theZakharov–Mikhailov–Burtsev integrable systems (Burtsev et al. 1987) withvariable spectral parameter, the same comments are indeed true. Here we haveto mention also important progress in computing the objects connected witharbitrary algebraic curves algorithmized by Deconinck and van Hoeij inMAPLE—see, for example, their paper (Deconinck & van Hoeij 2001). Furtherdevelopments involving calculations with Riemann theta functionsimplemented in Maple VIII and higher versions are described in a recentarticle by Deconinck et al. (2004). More detailed references on the computationof finite-gap solutions can be found in Deconinck & van Hoeij (2001),Deconinck et al. (2004), Frauendiener & Klein (2004, 2006) and Klein &Richter (2005). In particular, in Frauendiener & Klein (2004, 2006) andKlein & Richter (2005), the interested reader can find the algorithms forcomputing that are most rapid in the case of the hyperelliptic curves. The lateralgorithms using the MATLAB software allow the production of plots of solutionsto the Ernst equation, for which the modular parameters of the underlyingspectral curves depend on space and time variables.

Therefore, it is now clear that the finite-gap solutions have the same practicalvalue as standard special functions.

25 In my opinion, Burnside’s original text was better written compared with its expositionby Baker.26 For the KdV case, the realization of the same approach was much easier.



V. B. Matveev864


(b ) Deformations of Riemann surfaces in finite-gap integration

Here, we summarize some developments involving deformations of Riemannsurfaces and their application to solutions of physical and mathematicalproblems. Some of these developments were directly influenced by the methodof finite-gap integration. The relationship of other developments to finite-gapintegration was realized only a posteriori, confirming once more an impressiveuniversality of the ideas behind the method.

The first, and most physically important 1C1 differential equation, whose theta-functional solutions involve deformations of algebraic curve (Korotkin 1988;Korotkin & Matveev 1988, 1990) is the Ernst equation from general relativity,which can be written as follows in terms of a complex-valued function Eðz; rÞ:

ðEC �EÞ Ezz C1

rEr CErr

� �Z 2 E2

z CE2r

� �:

Theta-functional solutions of the Ernst equation can be written down in terms ofthe hyperelliptic ‘spectral curve’

w2 Z ðlKzKirÞðlKzC irÞY2gjZ1

ðlK ljÞ;

where lj are constants (immovable branch points) satisfying appropriate realityconditions. Two other branch points, zCir and zKir of the spectral curve depend onspace–time variables. The theta-functional solutions of Ernst equation lookas follows:

Eðr; zÞZQ

p

q

ðUðN1ÞKUðzC irÞÞ

Qp

q

ðUðN2ÞKUðzC irÞÞ

; ð2:1Þ

where p, q are two constant g-dimensional vectors also satisfying certain realityconditions;U is the Abel map; andN1,2 are the points at infinity of two sheets of thespectral curve. Dependence on the space variables enters equation (2.1) via thematrix of b-periods and the Abel map.

This drastically changes the properties of theta-functional solutions of thisequation as compared with solutions of KdV-like systems. In particular, thisclass of solutions contains a big supply of asymptotically flat solutions. In genus2, these solutions were successfully applied to solve the boundary-value problemscorresponding to an infinitely thin rotating dust disc (Neugebauer & Meinel1995; Klein & Richter 1998, 2005). Although, originally, solutions (2.1) werederived in Korotkin (1988) and Korotkin & Matveev (1988) using the frameworkof the Riemann–Hilbert problem, it was realized recently (Klein et al. 2002) thatthese solutions can be also derived directly from Fay’s bilinear identities if oneuses in addition Rauch’s variational formulae that describe the dependence ofholomorphic objects on Riemann surfaces on their moduli (Rauch 1959).

Similar solutions (Korotkin & Matveev 1990) can be constructed for otherintegrable systems belonging to the class of systems with ‘variable spectralparameter’ (Burtsev et al. 1987; Matveev 1994; Kokotov & Korotkin 2008).

A simple example of an integrable system in dimension 1, where modulardependence of theta-functions plays a principal role (which means that theunderlying ‘spectral curve’ is deforming), is given by the classical Halphen





system for three functions wj of variable m

vwj

vmZKwkwl Cwjðwk CwlÞ;

for i, j, kZ1, 2, 3. The solution of the system is given by logarithmic derivatives oftheta-constants: wkZ2ðd=duÞlog qkC1 for kZ1, 2, 3. The Halphen system togetherwith its solutions was used by Atiyah & Hitchin (1988) in their description of thetwo-monopole solution space. A generalization of the Halphen system,

vwj

vmZKwkwlK2wj

d

dmlogfqkC1qlC1g; ð2:2Þ

which is equivalent to SU(2) invariant self-dual Einstein’s equations, was shown byHitchin (1995) to be a special case of the Painleve 6 equation and solved in a rathersophisticated way. Later, simple formulae in terms of ‘theta-constants’ forcorresponding solutions were found by Babich & Korotkin (1998).

The possibility of solving the Ernst equation, the system (2.2), in terms oftheta-functions turns out to be the corollary of the explicit solvability of anarbitrary inverse monodromy problem with quasi-permutation N!N mono-dromy matrices in terms of the Szego kernel on Riemann surfaces (Korotkin2004); the simpler 2!2 case was solved earlier in Korotkin & Kitaev (1998) andDeift et al. (1999a). The new derivation of the solutions of the Ernst equation,based on the solution of this monodromy problem, was obtained by Korotkin &Matveev (1999, 2000).

Even earlier than the explicit deformations of algebraic curves describedabove, the rather sophisticated implicit deformations of spectral curves enteredthe scene. These deformations arise from the study of so-called Whithammodulations of theta-functional solutions of KdV-type system (Krichever 1988;Dubrovin 1992). The implicit deformations are described by a system ofWhitham equations on periods of some meromorphic differentials. Later, thisformalism was applied to the solution of integrable systems of hydrodynamictype by the so-called generalized hodograph method, see Dubrovin (1990) andKokotov & Korotkin (2008), and found an unexpected application the Seiberg–Witten theory of the low-energy effective action of supersymmetric gaugetheories (see Gorsky et al. (1995) for more details).

Another important area of applications of deformations of algebraic curves isDubrovin’s theory of Frobenius manifolds (Dubrovin 1996, 1999), which providesa geometrical formulation of Witten–Dijkgraaf–Verlinde–Verlinde equationsfrom two-dimensional topological field theory. The same framework offlat potential diagonal metrics, which forms the basis of the theory of systemsof hydrodynamic type, provides the geometrical background of the theory ofFrobenius manifolds. The most well-studied class of Frobenius manifolds isrelated to Hurwitz spaces, i.e. the spaces of deformations of Riemann surfacesrepresented as branched coverings of the Riemann sphere (Dubrovin 1996;Shramchenko 2005).

Finally, we would like to mention the massive interference of the algebraiccurves and methods of algebraic geometry with physics related to thedevelopment of perturbative string theory in the mid-1980s. After 1986, thetheta functions, Abelian integrals and Teichmuller spaces became standard tools



V. B. Matveev866


in the community of high-energy theoretical physicists due to the works byAlvarez-Gaume et al. (1986), Beilinson & Manin (1986), Belavin & Knizhnik(1986), Knizhnik (1987) and many others.

It is worthwhile mentioning that the same objects, spectral curves, Abelianintegrals, theta-functions, and more specifically the finite-gap integration method,play an increasing role in the theory of matrix models (e.g. Beisert et al. 2005, 2006;Eynard et al. 2005; Chekhov 2006).

I think that implicitly the experience of applying the algebro-geometricmethods to integrable systems played an important role for the stringy activityfrom the very beginning. The precise connections are still far from completelyexplored. As an example, we mention the relationship between the isomono-dromic tau-function, the G-function of Frobenius manifolds and determinants ofLaplacians over Riemann surfaces established in recent work (Kokotov & Korotkin2004). The reader can find many references in Kaku (1998) and some first linksbetween quantum strings and finite-gap integration topics in Saito (1987). (See alsovan Moerbeke (1994) for a comprehensive review.)

Finally, we would like to mention the works by Smirnov (1993, 1994, 2000a,b)where the role of the finite-gap sector in quantizing integrable models and inanalysing the quasi-classical limit of conformal field theory has been pointed outexplicitly.

(c ) Some other important developments

(i) Integrable tops and other finite-dimensional integrable systems

A few examples of finite-dimensional dynamical systems integrable in terms oftwo- or multi-dimensional Riemann theta functions were known already at the endof the nineteenth century and the beginning of the twentieth century. The mostfamous examples are connected with the works by Kowalyewsky, C. Neumann,Klebsch, Schottky, Steklov, Garnier and some other researchers. The developmentof the finite-gap integration method provided many new remarkable integrablecases as well as the systematization, and certain improvements, of the classicalresults. The references concerning this development can be found in various books(Bogoyavlenskij 1991; Belokolos et al. 1994; Reyman & Semenov-Tyan-Shansky2003) including, in particular, important results by Bobenko, Kuznetzov, vanMoerbeke and many other authors.

(ii) Integrable surfaces

The differential geometry of integrable surfaces represents another field, wherethe methods of finite-gap integration appear to be very powerful. It was stimulatedby the discovery of Wente of the constant mean curvature tori which contradict afamous Hopf conjecture, claiming that a sphere is the only possible compactconstant mean curvature surface inR3. Later works by Bobenko, Pinkall and someother authors (see Babich & Bobenko (1993) and Bobenko (1993) for references)provided further progress in this direction which allowed, in particular, the findingof a complete explicit description of the finite-gap Willmore tori in certain cases.Some, again incomplete, reviews of the related development also concerning thefinite-gap theory of the integrable discrete surfaces can be found in Bobenko &Pinkall (1999) together with further references.





(iii) Phase transitions and Peierls–Frolich models

It turns out that the finite-gap potentials corresponding to the Schrodingeroperator and its discrete version are connected with important problems in thetheory of quasi one-dimensional conductivity, namely with Peierls–Frolich phasetransitions. In particular, one-gap periodic potentials realize a minimum of thePeierls free energy functional, which was first proved by Belokolos in 1980 (seeBelokolos et al. (1994) for further references, and also Krichever (1982) for thereview of results connected with the discrete Peierls models studied by Kricheverand Dzyaloshinskij).

(iv) Finite-gap potentials and asymptotic problems of mathematical physics

There are numerous and quite non-trivial applications of formulae andmethods of finite-gap integration to various asymptotic problems of mathemat-ical physics, including some applications to the Schrodinger operator withdecreasing potential. Here we mention only the works by Dobrokhotov & Maslov(1980), Deift et al. (1997, 1999b), Kamvissis (1997), Kriecherbauer & Remling(2001) and Kamvissis et al. (2003). In Kamvissis (1997), it was shown how to getthe Dyson formula, providing the Fredholm determinant solution for the inversescattering problem, from the Its–Matveev formula.

(v) Infinite-dimensional theta functions and related solutions to the KdVand KP equation

There exists an infinite-dimensional generalization of the formula (1.23). On thistopic, the reader can consult articles (Muller et al. 1998; Schmidt 1996) where theimportant convergence problems are thoroughly analysed. Muller et al. (1998)contains also the references on important earlier works byMcKean and Trubowitz,Levitan and Knorrer concerning the hyperelliptic Riemann surfaces of infinitegenus. A similar extension of Krichever’s solution (1.28) of the KP equation to theinfinite genus case forms the content of a very recent book (Feldman et al. 2003).Unfortunately, in the infinite-dimensional genus case, the theta-functionalformulae are indeed less efficient, and their practical applications, contrary tothe finite genus case, are still missing for the moment.

(vi) Perturbations of finite-gap solutions and KAM theory

Here we mention only two recent books (Kappeler & Poschel 2003; Kuksin2000) that are completely devoted to the topic of this subsection whereadditional literature including references on the important contributions ofKuksin, Bikbaev, Bobenko and many other researchers can be found.

(vii) Application of the finite-gap integration in the theory of nonlinearocean waves

The interested reader can find the relevant information in the review article ofOsborn (2002), where the interesting application of finite-gap solutions of KdV,NLS and KP equations can be found.



V. B. Matveev868


3. Conclusion

I hope that even this brief and incomplete description of the evolution of thefinite-gap integration method and its links with different branches ofmathematics and physics, presented above, show that after 30 years of intensivedevelopment this area is still a very important and progressing branch of modernmathematical physics. Around this method were formed scientific schools inMoscow, New York and St Petersburg (though now dispersed across the differentcontinents). Later, the activity of these schools was extended by numerousindividual researchers all over the world. I hope that this volume will alsocontribute to further extensions of the method of finite-gap integration and itsapplications. Concluding I wish to thank Alexander Its and Dmitry Korotkin formany valuable comments on the draft of this article, and my wife Nina Matveevafor useful critical remarks.

This work was supported by grant ANR-05-BLAN-0029-01.

References

Ablowitz, M. J., Kaup, D. J., Newell, A. C. & Segur, H. 1973 Method for solving the sine-Gordonequation. Phys. Rev. Lett. 30, 1262–1264. (doi:10.1103/PhysRevLett.30.1262)

Airault, H., McKean, H. & Moser, J. 1977 Rational and elliptic solutions of the Korteweg–de Vriesequation and the related many body problems. Commun. Pure Appl. Math. 30, 95–148. (doi:10.1002/cpa.3160300106)

Akhiezer, N. I. 1960 Orthogonal polynomials on several intervals. Soviet Math. Dokl. 1, 989–992.Akhiezer, N. I. 1961 Continuous analogues of the polynomials orthogonal on the system of

intervals. Dokl. Akad. Nauk SSSR 141, 263–266.Alvarez-Gaume, L., Moore, G. & Vafa, C. 1986 Theta-functions, modular invariance, and strings.

Commun. Math. Phys. 106, 1–40. (doi:10.1007/BF01210925)Arbarello, E. & de Concini, C. 1987 Another proof of a conjecture of S. P. Novikov on periods of

Abelian integrals on Riemann surfaces. Duke Math. J. 54, 163–178. (doi:10.1215/S0012-7094-87-05412-3)

Atiyah, M. & Hitchin, N. 1988 The geometry and dynamics of magnetic monopoles. Princeton, NJ:Princeton University Press.

Babich, M. V. & Bobenko, A. I. 1993 Willmore tori with umbilic lines and minimal surfaces inhyperbolic space. Duke Math. J. 72, 151–186. (doi:10.1215/S0012-7094-93-07207-9)

Babich, M. V. & Bordag, L. A. 1985 Symmetric effectivization of three gap solutions of sine-Gordon equation and sinh-Gordon equation and their graphical representations. Communi-cations of JINR, P5-85-204, Dubna. [In Russian.]

Babich, M. V. & Bordag, L. A. 2005 Quasi periodic vortex structures in two-dimensional flows inan inviscid incompressible fluid. Russian J. Math. Phys. 12, 121–156.

Babich, M. & Korotkin, D. 1998 Self-dual SU(2)-invariant Einstein manifolds and modulardependence of theta-functions. Lett. Math. Phys. 46, 323–337. (doi:10.1023/A:1007542422413)

Babich, M. V., Bobenko, A. I. & Matveev, V. B. 1993 Reduction of Riemann theta functions of thegenus g to the theta functions of lower genera, and symmetries of algebraic curves. Dokl. Akad.Nauk. SSSR 272, 13–17. [In Russian.] [English transl. Soviet Math. Doklady 1993 28.]

Babich, M. V., Bobenko, A. I. & Matveev, V. B. 1985 Solutions of nonlinear equations in thetafunctions and symmetries of algebraic curves. Izv. Akad. Nauk SSSR Ser. Math. 49, 511–529.[In Russian.] [English transl. Math-Izv. Akad. Nauk SSSR Ser. Math. 1986 26.]

Baker, H. F. 1897 In Abel’s theorem and the allied aheorie of theta functions (ed. H. F. Baker),Cambridge, UK: Cambridge University Press.

Baker, H. F. 1907 Multiply periodic functions. Cambridge, UK: Cambridge University Press.


http://dx.doi.org/doi:10.1103/PhysRevLett.30.1262

http://dx.doi.org/doi:10.1002/cpa.3160300106


http://dx.doi.org/doi:10.1007/BF01210925

http://dx.doi.org/doi:10.1215/S0012-7094-87-05412-3



http://dx.doi.org/doi:10.1023/A:1007542422413




Baker, H. F. 1928 Note on the foregoing paper ‘Commutative ordinary differential operators’. Proc.

R. Soc. A 118, 584–593. (doi:10.1098/rspa.1928.0070)

Baxter, R. 2002 A rapidity-independent parameter in the star-triangle relation. In Mathphys

Odyssey 2001: integrable models and beyond: in honor of Barry M. McCoy (eds M. Kashivara &

T. Miwa), pp. 49–61. Boston, MA: Birkhauser.

Baxter, R. J., Perk, J. H. H. & Au-Yan, H. 1988 New solutions of the star-triangle relations for the

chiral Potts model. Phys. Lett. A 128, 138–142. (doi:10.1016/0375-9601(88)90896-1)

Beilinson, A. A. & Manin, Yu. I. 1986 The Mumford form and the Polyakov measure in string

theory. Commun. Math. Phys. 107, 359–380. (doi:10.1007/BF01220994)

Beisert, N., Kazakov, V. A., Sakai, K. & Zarembo, K. 2005 The algebraic curve of classical

superstrings on AdS5!S5. Commun. Math. Phys. 263, 659–710. (doi:10.1007/s00220-006-1529-4)

Beisert, N., Kazakov, V. A. & Sakai, K. 2006 Algebraic curve for the SO(6) Sector of AdS/CFT.

Commun. Math. Phys. 263, 611–657. (doi:10.1007/s00220-005-1528-x)

Belavin, A. A. & Knizhnik, V. G. 1986 Alebraic geometry and the geometry of quantum strings.

Phys. Lett. B 168, 201–206. (doi:10.1016/0370-2693(86)90963-9)

Belokolos, E. D., Bobenko, A. I., Enol’skii, V. Z. & Matveev, V. B. 1986 Algebraic–geometric

principles of superposition of finite-zone solutions of integrable non-linear equations. Russian

Math. Surveys 41, 1–49. (doi:10.1070/RM1986v041n02ABEH003241)

Belokolos, E. D., Bobenko, A. I., Enolskij, V. Z., Its, A. R. & Matveev, V. B. 1994 Algebro–

geometric approach to nonlinear integrable equations. Springer series studies in nonlinear

dynamics. Berlin, Germany: Spinger.

Bobenko, A. I. 1983 Finite-gap solutions in the language of automorphic functions. Kadomtcev–

Petviashvily equation. Zapiski Nauch. Semin. LOMI 129, 5–16. [English transl. J. Soviet Math.

1985 29, no. 2.]

Bobenko, A. I. 1985 Real algebro-geometric solutions of Landau–Lifshitz equation in Prym theta-

functions. Funct. Anal. Appl. 19, 5–17. (doi:10.1007/BF01086019)

Bobenko, A.I 1993 Surfaces in terms 2!2 matrices. Old and new integrable cases. In Harmonic

maps and integrable systems, vol. 23 (eds A. P. Fordy & J. C. Wood). Aspects of mathematics,

p. 83. Braunschweig, Germany: Vieweg & Sohn.

Bobenko, A. I. & Bordag, L. A. 1987 Qualitative analysis of finite gap solutions of the KdV

equation with help of the automorphic approach. Zapiski Nauch. Semin. LOMI 165, 31–41.

[English transl. J. Soviet Math. 1990 50, no. 6.]

Bobenko, A. I. & Bordag, L. A. 1989 Periodic multiphase solutions of the KP equation. J. Phys. A:

Math. Gen. 22, 1259–1274. (doi:10.1088/0305-4470/22/9/016)

Bobenko, A. I. & Pinkall, U. 1999 Discretisation of surfaces and integrable systems. In Discrete

integrable geometry and physics (eds A. I. Bobenko & U. Seiler), pp. 3–58. New York, NY:

Oxford University Press.

Bogoyavlenskij, O. I. 1991 Overturning solitons. Moscow: Nauka. [In Russian.] [Transl. J. Math.

Sci. 1999 (New York) 94, 1177–1217.]

Brezhnev, Y. V. 2008 What does integrability of finite-gap or soliton potentials mean? Phil.

Trans. R. Soc. A 366, 923–945. (doi:10.1098/rsta.2007.2056)

Buchstaber, V. M., Enol’skii, V. Z. & Leykin, D. V. 1997 Kleinian functions, hyperelliptic

Jacobians and applications. In Reviews in mathematics and mathematical physics, vol. 10:2 (eds

S. P. Novikov & I. M. Krichever), pp. 1–125. London, UK: Gordon and Breach.

Burchnall, J. L. & Chaundy, T. W. 1923 Commutative ordinary differential operators. Proc. Lond.

Math. Soc. 21, 420–440. (doi:10.1112/plms/s2-21.1.420)

Burchnall, J. L. & Chaundy, T. W. 1928 Commutative ordinary differential operators. Proc. R.

Soc. A 118, 557–583. (doi:10.1098/rspa.1928.0069)

Burnside, W. 1890 On functions determined from their discontinuities and a certain form of

boundary condition. Proc. Lond. Math. Soc. 22, 346–358. (doi:10.1112/plms/s1-22.1.346)

Burnside, W. 1892 On a class of automorphic functions. Proc. Lond. Math. Soc. 23, 49–88. (doi:10.

1112/plms/s1-23.1.49)


http://dx.doi.org/doi:10.1098/rspa.1928.0070

http://dx.doi.org/doi:10.1016/0375-9601(88)90896-1


http://dx.doi.org/doi:10.1007/s00220-006-1529-4

http://dx.doi.org/doi:10.1007/s00220-005-1528-x

http://dx.doi.org/doi:10.1016/0370-2693(86)90963-9

http://dx.doi.org/doi:10.1070/RM1986v041n02ABEH003241


http://dx.doi.org/doi:10.1088/0305-4470/22/9/016

http://dx.doi.org/doi:10.1098/rsta.2007.2056

http://dx.doi.org/doi:10.1112/plms/s2-21.1.420

http://dx.doi.org/doi:10.1098/rspa.1928.0069





V. B. Matveev870


Burtsev, S. P., Mikhailov, A. V. & Zakharov, V. E. 1987 The inverse scattering method with

variable spectral parameter. Theor. Math. Phys. 70, 227–240. (doi:10.1007/BF01040999)Chekhov, L. O. 2006 Solving matrix models in the 1/N-expansion. Russian Math. Surveys 61,

93–156. (doi:10.1070/RM2006v061n03ABEH004329)Chen, Y. & Its, A. R. 2008 A Riemann–Hilbert approach to the Akhiezer polynomials. Phil. Trans.

R. Soc. A 366, 973–1003. (doi:10.1098/rsta.2007.2058)Cherednik, I. 1996 Basic methods of soliton theory, vol. 25. Advanced series in mathematical

physics. River Edge, NJ: World Scientific.Darboux, G. 1882a Sur une proposition relative aux equations lineaires. C. R. Acad. Sci. Paris 94,

1456–1459.Darboux, G. 1882b Sur une equation lineaire. C. R. Acad. Sci. Paris 94, 1645–1648.Darboux, G. 1894 Lecons sur la theorie tenerale des surfaces et les applications geometriques du

calcul infinitesimal, vol. 1,2. Paris, France: Gauthier-Villars.Date, E. & Tanaka, S. 1976 Periodic multi-soliton solutions of Korteweg–de Vries equation and

Toda lattice. Progr. Theor. Phys. Suppl. 59, 107–128.Deconinck, B. & van Hoeij, M. 2001 Computing Riemann matrices of algebraic curves. Physica D

152/153, 28–46. (doi:10.1016/S0167-2789(01)00156-7)Deconinck, B., Heil, M., Bobenko, A., van Hoeij, M. & Shmies, M. 2004 Computing Riemann theta

functions. Math. Comp. 73, 1417–1442. (doi:10.1090/S0025-5718-03-01609-0)Deift, P., Its, A. & Zhou, X. 1997 A Riemann–Hilbert approach to asymptotic problems arising in

the theory of random matrix models, and also in the theory of integrable statistical mechanics.

Ann. Math. 146, 149–235. (doi:10.2307/2951834)Deift, P., Its, A., Kapaev, A. & Zhou, X. 1999a On the algebro–geometric integration of the

Schlesinger system and on elliptic solutions of the Painleve VI equation. Commun. Math. Phys.203, 613–633. (doi:10.1007/s002200050037)

Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakidies, S. & Zhou, X. 1999b Uniform

asymptotics for polynomials orthogonal with respect to varying exponential weights andapplications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52,1335–1425. (doi:10.1002/(SICI)1097-0312(199911)52:11!1335::AID-CPA1O3.0.CO;2-1)

Dickey, L. A. 1955 The zeta function of an ordinary differential equation on a finite interval. Izv.

Akad. Nauk SSSR. Ser. Mat. 19, 187–200.Dickey, L. A. 1991 Soliton equations and Hamiltonian systems. Advanced series in mathematical

physics. Singapore: World Scientific.Dirac, P. A. M. 1977 Recollection of an exciting era. In History of twentieth century physics. Proc.

International School of Physics Enrico Fermi, 57, pp. 109–146. New York, NY: Academic Press.Dobrokhotov, S. Yu. & Maslov, V. P. 1980 Finite-zone almost periodic solutions in WKB-

approximations. In Modern problems in mathematics, vol. 15 (Russian), pp. 3–94. Moscow,

Russia: Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii.Drach, J. 1919 Sur l’integration par quadratures de l’equation differentiel y00Z[q(x)Ch]y. Compt.

Rend. Acad. Sci. 168, 337–340.Dubrovin, B. A. 1975 The inverse scattering problem for periodic finite-zone potentials. Funct.

Anal. Appl. 9, 61–62. (doi:10.1007/BF01078183)Dubrovin, B. A. 1976 Finite-zone linear operators and Abelian varieties. Russian Math. Surveys 31,

259–260. (doi:10.1070/RM1976v031n01ABEH001446)Dubrovin, B. A. 1981 Theta functions and nonlinear equations. Russian Math. Surveys 36, 12–78.

(doi:10.1070/RM1981v036n02ABEH002596)Dubrovin, B. A. 1990 On the differential geometry of strongly integrable systems of hydrodynamics

type. Funct. Anal. Appl. 24, 280–285. (doi:10.1007/BF01077332)Dubrovin, B. A. 1992 Hamiltonian formalism of Whitham-type hierarchies and topological

Landau–Ginzburg models. Commun. Math. Phys. 145, 195–207. (doi:10.1007/BF02099286)Dubrovin, B. 1996 Geometry of 2D topological field theories. In Integrable systems and quantum

groups (Montecatini Terme, 1993). Lecture notes in mathematics, pp. 120–348. Berlin,Germany: Springer.





http://dx.doi.org/doi:10.1016/S0167-2789(01)00156-7


http://dx.doi.org/doi:10.2307/2951834

http://dx.doi.org/doi:10.1007/s002200050037

http://dx.doi.org/doi:10.1002/(SICI)1097-0312(199911)52:11%3C1335::AID-CPA1%3E3.0.CO;2-1

http://dx.doi.org/doi:10.1002/(SICI)1097-0312(199911)52:11%3C1335::AID-CPA1%3E3.0.CO;2-1









Dubrovin, B. 1999 Painleve transendents in two-dimensional topological field theory. In The

Painleve property CRM Ser. Math. Phys., pp. 287–412. New York, NY: Springer.Dubrovin, B. A. & Natanzon, S. M. 1989 The real nonsingular solutions of the KP equation. Math.

USSR—Izv. 32, 269–288. (doi:10.1070/IM1989v032n02ABEH000759)Dubrovin, B. A. & Novikov, S. P. 1974 Periodic and conditionally periodic analogs of the many-

soliton solutions of the Korteweg–de Vries equation. Soviet Physics JETP 40, 1058–1063.Dubrovin, B. A., Matveev, V. B. & Novikov, S. P. 1976 Nonlinear equations of Korteweg–de Vries

type, finite-zone linear operators and Abelian varieties. Russian Math. Surveys 31, 56–134.(doi:10.1070/RM1976v031n01ABEH001446)

Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G. 1955 Higher transcendental functionsBased, in part, on notes left by Harry Bateman, vol. III. New York, NY: McGraw-Hill.

Eynard, B., Kokotov, A. & Korotkin, D. 2005 1/N 2 correction to free enerrgy in Hermitian two-matrix model. Lett. Math. Phys. 71, 199–207. (doi:10.1007/s11005-005-0157-9)

Feldman, J., Knorrer, H. & Trubowitz, E. 2003 Riemann surfaces of infinite genus. CRM

monograph series. Providence, RI: American Mathematical Society.Flaschka, H. & McLaughlin, D. W. 1976 Canonically congugate variables for the Korteweg–de

Vries Equation and Toda lattice with periodic boundary conditions. Prog. Theoret. Phys. 55,438–456. (doi:10.1143/PTP.55.438)

Frauendiener, J. & Klein, C. 2001 Exact Relativistic treatment of stationary counterrotating dustdiscs: physical properties. Phys. Rev. D 63, 084 025. (doi:10.1103/PhysRevD.63.084025)

Frauendiener, J. & Klein, C. 2004 Hyperelliptic theta-functions and spectral methods. J. Comput.

Appl. Math. 167, 193–218. (doi:10.1016/j.cam.2003.10.003)Frauendiener, J. & Klein, C. 2006 Hyperelliptic theta-functions and spectral methods: KdV and

KP Solutions. Lett. Math. Phys. 76, 249–267. (doi:10.1007/s11005-006-0068-4)Gardner, C. S., Green, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving KdV

equation. Phys. Rev. Lett. 15, 1095–1097. (doi:10.1103/PhysRevLett.19.1095)Gelfand, I. M. & Dickey, L. A. 1979 Integrable nonlinear equations and the Liouville theorem.

Funct. Anal. Appl. 13, 6–15. (doi:10.1007/BF01076434)Gesztesy, F. & Holden, H. 2003 Soliton equations and their algebro-geometric solutions: (1C1)-

dimensional continuous models. Cambridge studies in advanced mathematics, vol. 1.Cambridge, UK: Cambridge University Press no. 79

Glazman, I. M. 1963 Direct methods of qualitative spectral analysis of singular differentialoperators. Moscow, Russia: Fizmatgiz. [In Russian.]

Gorsky, A., Krichever, I., Marshakov, A., Mironov, A. & Morozov, A. 1995 Integrability andSeiberg–Witten exact solution. Phys. Lett. B 355, 466–474. (doi:10.1016/0370-2693(95)00723-X)

Hitchin, N. 1995 Twistor spaces. Einstein metrics and isomonodromic deformations. J. Diff. Geom.

42, 30–112.Hochstadt, H. 1965 On the determination of a Hill’s equation from its spectrum. Arch. Rat. Mech.

Anal. 19, 353–362. (doi:10.1007/BF00253484)Ince, E. L. 1940a The periodic Lame functions. Proc. R. Soc. Edin. 60, 47–63.Ince, E. L. 1940b Further investigations into the periodic Lame functions. Proc. R. Soc. Edin. 60,

83–99.Its, A. R. 1976 Inversion of hyperelliptic integrals and integration of nonlinear differential

equations. Vestnik Leningrad Univ. Math. 9, 121–128.Its, A. R. & Kotlyarov, V. P. 1976 Explicit formulas for solutions of a nonlinear Schrodinger

equation. Dokl. Akad. Nauk Ukrain. SSR Ser. 11, 965–968.Its, A. R. & Matveev, V. B. 1975a Hill operators with a finite number of lacunae. Funct. Anal.

Appl. 9, 65–66. (doi:10.1007/BF01078185)Its, A. R. & Matveev, V. B. 1975b Schrodinger operators with the finite-band spectrum and the

N-soliton solutions of the Korteweg–de Vries equation. Theor. Math. Phys. 23, 343–355. (doi:10.1007/BF01038218)

Its, A. R. & Matveev, V. B. 1976 On a class of solutions of the KdV equations. In ProblemyMatematicheskoj Physiki, no. 8 (ed. M. Sh. Birman). Leningrad University. [In Russian.]


http://dx.doi.org/doi:10.1070/IM1989v032n02ABEH000759



http://dx.doi.org/doi:10.1143/PTP.55.438

http://dx.doi.org/doi:10.1103/PhysRevD.63.084025

http://dx.doi.org/doi:10.1016/j.cam.2003.10.003




http://dx.doi.org/doi:10.1016/0370-2693(95)00723-X






V. B. Matveev872


Its, A. R. & Matveev, V. B. 1984 Algebro-geometrical integration of the modified nonlinear

Schrodinger equation, finite-gap solutions and their degenerations. J. Soviet Math. 21, 64–77.Kac, M. & van Moerbeke, P. 1975 On some periodic Toda lattices. Proc. Natl Acad. Sci. USA 72,

1627–1629. (doi:10.1073/pnas.72.4.1627)Kaku, M. 1998 Introduction to superstrings. New York, NY: Springer.Kamvissis, S. 1997 Inverse scattering as an infinite period limit. C. R. Acad. Sci. Paris Ser. I Math.

325, 969–974.Kamvissis, S., McLaughlin, K. T.-R. & Miller, P. D. 2003 Semiclassical soliton ensembles for the

focusing nonlinear Schrodinger equation. Annals of mathematical studies, vol. 154. Princeton,

NJ: Princeton University Press.Kappeler, T. & Poschel, J. 2003 KdV KAM Results in mathematics and related areas. 3rd series,

vol. 45. Berlin, Germany: Springer.Khare, A. & Sukhatme, U. 2005 PT-invariant periodic potentials with a finite number of band

gaps. J. Math. Phys. 46, 082106. (doi:10.1063/1.2000207)Klein, C. & Richter, O. 1998 Explicit solutions of Riemann–Hilbert problem for the Ernst equation.

Phys. Rev. D 57, 857–862. (doi:10.1103/PhysRevD.57.857)Klein, C. & Richter, O. 2005 Ernst equation and Riemann surfaces: analytical and numerical

methods. Springer lecture notes in physics. Berlin, Germany: Springer.Klein, C., Korotkin, D. & Shramchenko, V. 2002 Ernst equation, Fay identities and variational

formulas on hyperelliptic curves. Math. Res. Lett. 9, 1–20.Klein, C., Matveev, V. B. & Smirnov, A. O. In press. Cylindrical Kadomtsev–Petviashvily

equation: old and new results. J. Theor. Math. Phys.Knizhnik, V. G. 1987 Analytic fields on the Riemann surfaces. II. Commun. Math. Phys. 112,

567–590. (doi:10.1007/BF01225373)Knorrer, H. 1982 Geodesics on quadrics and a mechanical problem of Neumann. J. Reine Angew.

Math. 334, 69–78.Kokotov, A. & Korotkin, D. 2004 Tau-functions on Hurwitz spaces. Math. Phys. Anal. Geom. 7,

47–96. (doi:10.1023/B:MPAG.0000022835.68838.56)Kokotov, A. & Korotkin, D. 2008 A new hierarchy of integrable systems associated to Hurwitz

spaces. Phil. Trans. R. Soc. A 366, 1055–1088. (doi:10.1098/rsta.2007.2061)Korotkin, D. A. 1988 Finite gap solutions of the stationary axisymmetric Einstein equation in

vacuum. Theor. Math. Phys. 77, 1018–1031. (doi:10.1007/BF01028676)Korotkin, D. A. 1996 On some integrable cases in surface theory. Zapiski Nauch. Semin. LOMI 234,

65–124.Korotkin, D. 2004 Solution of matrix Riemann–Hilbert problems with quasi-permutation

monodromy matrices. Math. Ann. 329, 335–364. (doi:10.1007/s00208-004-0528-z)Korotkin, D. A. & Kitaev, A. V. 1998 On solutions of the Schlesinger equations in terms of

Q-functions. Int. Math. Res. Notices 17, 877–905.Korotkin, D. A. & Matveev, V. B. 1988 Finite-gap integration of nonlinear differential

equations having U-V pairs with a variable spectral parameter. In Some topics on inverse

problems (ed. P. C. Sabatier), pp. 182–211. Singapore: World Scientific.Korotkin, D. A. & Matveev, V. B. 1990 Algebraic–geometrical solutions of gravitational equations.

Leningrad Math. J. 1, 379–408.Korotkin, D. A. & Matveev, V. B. 1999 Schlesinger system, Einstein equations and hyperelliptic

curves. Lett. Math. Phys. 49, 145–159. (doi:10.1023/A:1007626009053)Korotkin, D. A. & Matveev, V. B. 2000 Theta-functional solutions of the Schlesinger system and

the Ernst equation. Funct. Anal. Appl. 34, 252–262. (doi:10.1023/A:1004153222818)Kozel, V. A. & Kotlyarov, A. P. 1976 Almost periodic solutions of the sine-Gordon equation. Dokl.

Akad. Nauk Ukrain. SSR Ser. 10, 878–881.Krichever, I. M. 1975 Reflectionless potentials on the finite-gap background. Funct. Anal. Appl. 9,

84–87. (doi:10.1007/BF01075460)Krichever, I. M. 1976a An algebraic–geometrical construction of the Zakharov-Shabat equations

and their periodic solutions. Soviet Math. Dokl. 15, 394–397.


http://dx.doi.org/doi:10.1073/pnas.72.4.1627

http://dx.doi.org/doi:10.1063/1.2000207



http://dx.doi.org/doi:10.1023/B:MPAG.0000022835.68838.56



http://dx.doi.org/doi:10.1007/s00208-004-0528-z







Krichever, I. M. 1976b Algebraic curves and commuting matrix differential operators. Funct. Anal.

Appl. 10, 75–76. (doi:10.1007/BF01077946)Krichever, I. M. 1978 Algebraic curves and nonlinear difference operators. Russian Math. Surveys

33, 215–216. (doi:10.1070/RM1978v033n04ABEH002503)Krichever, I. M. 1980 Elliptic solutions of the Kadomtsev–Petviashvili equations and integrable

particle systems. Funct. Anal. Appl. 14, 282–290. (doi:10.1007/BF01078304)Krichever, I. M. 1981 Periodic non-abelian Toda lattice and its two-dimensional generalisations.

Appendix to Belokolos et al. (1994).Krichever, I. M. 1982 The Peierls model. Funct. Anal. Appl. 16, 248–263. (doi:10.1007/

BF01077847)Krichever, I. M. 1988 Method of averaging for two-dimensional ‘integrable’ equations. Funct. Anal.

Appl. 22, 200–213. (doi:10.1007/BF01077626)Krichever, I. M. & Novikov, S. P. 1980 Holomorphic vector bundles over algebraic curves and

nonlinear equations. Russian Math. Surveys 35, 47–68. (doi:10.1070/RM1980v035n06AB

EH001974)Krichever, I. M. & Novikov, S. P. 2003 Two-dimensional Toda lattice, commuting difference

operators and holomorphic bundles. Russian Math. Surveys 58, 50–88. (doi:10.1070/RM2003v058n03ABEH000628)

Kriecherbauer, T. & Remling, C. 2001 Finite gap potentials and WKB asymptotics for one-dimensional Schrodinger operators. Commun. Math. Phys. 223, 409–435. (doi:10.1007/

s002200100550)Kuksin, S. B. 2000 Analysis of Hamiltonian PDEs, vol. 19. Oxford lecture series in mathematics

and its applications. Oxford, UK: Oxford University Press.Lax, P. D. 1968 Integrals of nonlinear equation of evolution and solitary waves. Commun. Pure

Appl. Math. 21, 467–490. (doi:10.1002/cpa.3160210503)Lax, P. D. 1975 Periodic solutions of the KdV equation. Commun. Pure Appl. Math. 28, 141–148.Lazutkin, V. F. & Pankratova, T. F. 1975 Normal forms and versal deformations for Hill’s

equation. Funct. Anal. Appl. 9, 306–311. (doi:10.1007/BF01075876)Lipowskij, V. D., Matveev, V. B. & Smirnov, A. O. 1986 On the relation between the KP equation

and Johnson equation. Zapiski Nauch. Semin. LOMI 46, 70–75. Transl. J. Sov. Math.Magnus, W. & Winkler, S. 1979 Hill’s equation. New York, NY: Dover.Marchenko, V. A. 1974 Periodic problem of Korteweg de Vries Equation I. Matem. Sbornik 95,

331–356. [In Russian.]Marchenko, V. A. 1977 Sturm–Liouville operators and their applications. Kiev, Ukraine: Naukova

Dumka. [In Russian.]Marchenko, V. A. & Ostrovskij, I. V. 1975 Characterization of spectrum of Hill’s operator. Matem.

Sbornik 97, 540–586.Matveev, V. B. 1975 New scheme of integration of the Korteweg de Vries equation (Lecture given

at the Petrovskij seminar of 26 March 1975 in Moscow). Uspekhi Matem. Nauk 30, 201–203.Matveev, V. B. 1976 Abelian functions and solitons. Wroclaw, Poland: Wroclaw University.

Preprint no. 373.Matveev, V. B. 1979a Darboux transformation and explicit solutions of the Kadomtcev–

Petviaschvily equation, depending on functional parameters. Lett. Math. Phys. 3, 213–216.(doi:10.1007/BF00405295)

Matveev, V. B. 1979b Darboux transformations and explicit solutions of differential-difference anddifference-difference evolution equations. Lett. Math. Phys. 3, 217–222. (doi:10.1007/

BF00405296)Matveev, V. B. 1994 Deformations of algebraic curves and integrable nonlinear evolution

equations. Acta Applicanda Math. 36, 1–24. (doi:10.1007/BF01001538)Matveev, V. B. 2002 Positons: slowly decreasing analogues of solitons. Theor. Math. Phys. 131,

483–497. (doi:10.1023/A:1015149618529)Matveev, V. B. & Salle, M. A. 1991 Darboux transformations and solitons. Springer series in

nonlinear dynamics. Berlin, Germany: Springer.






















V. B. Matveev874


Matveev, V. B. & Smirnov, A. O. 1990 Some comments on the solvable chiral Potts model. Lett.

Math. Phys. 19, 179–185. (doi:10.1007/BF01039310)

Matveev, V. B. & Smirnov, A. O. 1993 Symmetric reductions of the Riemann theta function and

some applications to the nonlinear Schrodinger and Boussinesq equation. Amer. Math. Soc.

Transl. Ser. 157, 227–237.

Matveev, V. B. & Smirnov, A. O. 2006 On the link between the Sparr equation and Darboux–

Verdier–Treibich equation and Sparr equation. Lett. Math. Phys. 76, 283–285. (doi:10.1007/

s11005-006-0074-6)

Matveev, V. B. & Yavor, M. A. 1979 Solutions presque periodiques et a N-solitons de l’equation

hydrodynamique non lineaire de Kaup. Ann. Inst. H. Poincare, Sect. A (N.S.) 31, 25–41.

McKean, H. P. & van Moerbeke, P. 1975 The spectrum of Hill’s equation. Invent. Math. 30,

217–274. (doi:10.1007/BF01425567)

Muller, W., Schmidt, M. & Schrader, R. 1998 Hyperelliptic Riemann surfaces of infinite genus and

solutions of the KDV equation. Duke Math. J. 91, 315–352. (doi:10.1215/S0012-7094-98-09114-1)

Mumford, D. 1983 Tata lectures on theta, vol. I, II. Boston, MA: Birkhauser.

Mumford, D. & van Moerbeke, P. 1979 Spectrum of difference operators and algebraic curve. Acta

Math. 143, 93–154. (doi:10.1007/BF02392090)

Naiman, P. B. 1962 On the theory of periodic and limit-periodic Jacobian matrices. Soviet Math.

Doklady 3, 383–385.

Neugebauer, G. & Meinel, R. 1995 General relativistic gravitational field of the rigidly rotating

disk of dust: solution in terms of ultraelliptic functions. Phys. Rev. Lett. 75, 3046–3048. (doi:10.

1103/PhysRevLett.75.3046)

Novikov, S. P. 1974 A periodic problem for the Korteweg–de Vries equation. I. Funct. Anal. Appl.

8, 236–246. (doi:10.1007/BF01075697)

Novikov, S. P. & Dynnikov, I. 1997 Discrete spectral symmetries of low dimensional differential

operators and difference operators on regular lattices. Russian Math. Surveys 52, 1057–1116.

(doi:10.1070/RM1997v052n05ABEH002105)

Novikov, S. P., Manakov, S. V., Pitayevskij, L. P. & Zakharov, V. E. 1984 Theory of solitons. The

inverse scattering method. Contemporary Soviet mathematics. New York, NY: Consultants

Bureau.

Osborn, A. R. 2002 Nonlinear ocean waves and the inverse scattering transform. In Scattering,

vol. I (eds R. Pike & P. C. Sabatier), pp. 637–665. London, UK: Academic Press.

Rauch, H. E. 1959 Weierstrass points, branch points, and moduli of Riemann surfaces. Commun.

Pure Appl. Math. 12, 543–560. (doi:10.1002/cpa.3160120310)

Reyman, A. G. & Semenov-Tyan-Shansky, M. A. 2003 Integrable systems: group theoretic

approach. Moscow-Izhevsk, Russia: Institute of Computer Studies. [In Russian.]

Saito, S. 1987 String amplitudes as solutions to soliton equations. Phys. Rev. D 36, 1819–1826.

(doi:10.1103/PhysRevD.36.1819)

Schmidt, M. U. 1996 Integrable systems and Riemann surfaces of infinite genus. Memoirs of the

American Mathematical Society, vol. 122. Providence, RI: American Mathematical Society.

Shabat, A. B. & Veselov, A. V. 1993 Dressing chains and the spectral theory of the Schrodinger

operator. Funct. Anal. Appl. 27, 81–96. (doi:10.1007/BF01085979)

Shiota, T. 1986 Characterisation of Jacobian varieties in terms of soliton equations. Invent. Math.

83, 333–382. (doi:10.1007/BF01388967)

Shramchenko, V. 2005 “Real doubles” of Hurwitz Frobenius manifolds. Commun. Math. Phys. 256,

635–680. (doi:10.1007/s00220-005-1321-x)

Smirnov, F. A. 1993 Form factors, deformed Knizhnik-Zamolodchikov equations, and finite-gap

integration. Commun. Math. Phys. 155, 459–487. (doi:10.1007/BF02096723)

Smirnov, A. O. 1994 Finite gap elliptic solutions of the KdV equations. Acta Appl. Math. 3,

105–140.

Smirnov, F. A. 1994 What are we quantizing in integrable models of field theory? Algebra i Analiz

6, 248–261. [Transl. St.-Petersburg Math. J. 1995 6, 417–428.]
















http://dx.doi.org/doi:10.1007/s00220-005-1321-x





Smirnov, F. A. 2000a Quasi-classical study of form factors in finite volume. In L. D. Faddeev’sseminar on mathematical physics, vol. 201 (eds M. Semenov-Tian-Shansky). AMS translationsseries 2, pp. 283–307. Providence, RI: American Mathematical Society.

Smirnov, F. A. 2000b Dual Baxter equations and quantization of the affine Jacobian. J. Phys. AMath. Gen. 33, 3385–3405. (doi:10.1088/0305-4470/33/16/323)

Smirnov, A. O. 2002 Elliptic solitons and Heun’s equation. In The Kowalewski property (Leeds,2001). CRM Proc. lecture notes, pp. 287–305. Providence, RI: American Mathematical Society.

Takasaki, K. 2003 Spectral curve, Darboux coordinates and Hamiltonian structure of periodicdressing chains. Commun. Math. Phys. 241, 111–142. (doi:10.1007/s00220-003-0929-y)

Titchmarsh, E. C. 1958 Eigenfunction expansions associated with second order differentialequations, vol. 2. Oxford, UK: Clarendon Press.

Treibich, A. 1989 Tangential polynomials and elliptic solitons. Duke Math. J. 59, 611–627. (doi:10.1215/S0012-7094-89-05928-0)

van Moerbeke, P. 1994 Integrable foundations of string theory. In Lectures on integrable systems.In memory of Jean-Louis Verdier. Proc. CIMPA School on Integrable Systems, Sophia-Antipolis, 10–28 June 1991 (eds O. Babelon, P. Cartier & Y. Kosmann-Schwarzbach),pp. 163–267. River Edge, NJ: World Scientific.

Veselov, A. 1980 Finite-zone potentials and integrable systems on the sphere with quadraticpotentials. Funct. Anal. Appl. 14, 37–39. (doi:10.1007/BF01078414)

Wilson, G. 1981 Integrable systems. Selected papers. London mathematical society lecture noteseries. Cambridge, UK: Cambridge University Press.

Witten, E. 1988 Coadjoint orbits of the Virasoro group. Commun. Math. Phys. 114, 1–53. (doi:10.1007/BF01218287)

Zakharov, V. E. & Faddeev, L. D. 1971 The Korteweg–de Vries equation is a fully integrableHamiltonian system. Funct. Anal. Appl. 5, 280–287. (doi:10.1007/BF01086739)

Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of the 2D self focusing and 1D auto modulationin nonlinear media. Soviet Physics JETP 34, 62–69.

Zverovich, E. I. 1971 Boundary value problems for analytical functions in the Holder classes on theRiemann surfaces. Russian Math. Surveys 26, 114–179. (doi:10.1070/rm1971v026n01ABEH003811)


http://dx.doi.org/doi:10.1088/0305-4470/33/16/323

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