NONLINEAR PROBLEMS IN CLASSICAL AND QUANTUMELECTRODYNAMICS

A.O. Barut

The Univers i ty of Colorado, Boulder, Colorado 80309

Table of Contents

Page

I . Introduct ion 2

I I . Classical R e l a t i v i s t i c Electron Theory 2

I I i o Quantum Theory of Se l f - l n te rac t ion 5

Other Remarkable Solutions of Nonlinear Equations

Some Related Problems

References

10

11

13

NONLINEAR PROBLEMS IN CLASSICAL AND QUANTUMELECTRODYNAMICS

A. O. Barut

The Univers i ty of Colorado, Boulder, Colorado 80309

I . INTRODUCTION

We present here a discussions of the non- l inear problems ar is ing due to se l f -

f i e l d of the e lectron, both in c lassical and quantum electrodynamics. Because of

some shortcomings of the conventional quantumelectrodynamics 11[ an attempt has

been made to carry over the nonperturbative rad ia t ion react ion theory of c lassical

electrodynamics to quantum theory. The goal is to have an equation for the rad ia t ing

and se l f - i n t e rac t i ng electron as a whole, in other words,an equation for the f i na l

"dressed" electron. In addi t ion the theory and renormalizat ion terms are a l l f i n i t e .

Each pa r t i c l e is described by a s ingle wave funct ion ~(x) moving under the inf luence

of the s e l f - f i e l d as well as the f i e l d of a l l other par t i c les . In pa r t i cu la r , we dis

cuss the completely covariant two-body equations in some de ta i l , and point out to

some new remarkable solut ions of the non- l inear equations: These are the resonance

states in the two-body problem due to the in terac t ion of the anomalous magnetic mo-

ment of the par t i c le which become very strong at small distances.

I I . CLASSICAL RELATIVISTIC ELECTRON THEORY

The motion of charged par t ic les are not governed by the simple set of Newton's

equations as one usual ly assumes in the theory of dynamical systems, but by rather

complicated non- l inear equations invo lv ing even th i rd order of der ivat ives. To see

th is we begin with Lorentz's fundamental postulates of the electron theory of mat-

te r :

( i ) Matter consists of a number of charged par t ic les moving under the inf luence

of the electromagnetic f i e l d produced by a l l charged par t i c les . The equation of mo-

t ion of the i th charged par t i c le is given by

m Z ( i ) : e F (x) Z V( i ) l (1) lx=z ( i ) '

where Z (S) is the worl 'd- l ine of the par t i c le in the Minkowski space M 4 in terms of

an invar ian t time parameter S (e.g. proper time) - the der ivat ives are with respect

to S, and F is the to ta l electromagnetic f i e l d . ~v

( i i ) The to ta l electromagnetic f i e l d F obeys Maxwell's equations P~

F '~(x) = j (x) , (2)

where j (x) is the tota l current of a l l the charges. For point charges we have

j~(x) = ~ e (k) ~(k)~ ~(x - Z (k))

We have in p r inc ip le a closed system of equations i f we have in addi t ion some

model of matter t e l l i n g us how many charged par t ic les there are.

(3)

These equations taken together give for each par t i c le i a h igh ly nonl inear

equation because due to the term k = i in (3), F (x) in (1) depends nonl inear ly

on Z ( i ) . This is the socalled s e l f - f i e l d of the Uith pa r t i c le . ~ Actua l ly th is term

is even i n f i n i t e at X = Z ( i ) due to the factor ~(X-Zki)). ' " In pract ice th is i n f i n i t e

term does not cause as much trouble as i t should-one simply drops such terms in

f i r s t approximation. The reason for th is is that a major part of the s e l f - f i e l d is

already taken into account as the i ne r t i a or mass of the par t i c le on the l e f t hand

side of e q . ( i ) : in other words, the mass m in ( I ) is the socalled renormalized

mass m R as I shal l explain in more de ta i l . Unfortunately not the whole of the s e l f -

f i e l d is an i n e r t i a l term in the presence of external forces. Otherwise the whole

electrodynamics would be a closed and consistent theory wi thout i n f i n i t e s . For a

s ingle pa r t i c l e , i t is true by d e f i n i t i o n , that a l l the s e l f - f i e l d is in the form

of an i n e r t i a l term because then the equation is mR~ , = O. But the presence of

other par t ic les modifies the cont r ibut ion of the s e l f - f i e l d to an i n e r t i a l term

mRZ. And th is is r ea l l y the whole story and problem of electrodynamics, c lassical

or quantummechanical: How much of the s e l f - f i e l d is ine r t ia? . Af ter the i n e r t i a l

term has been subtracted, the remainnder gives r ise to observable ef fects which we

ca l l rad ia t ive phenomena l i ke anomalous magnetic moment, Lamb s h i f t , etc. I w i l l

now show f i r s t how th is is done in c lassical electrodynamics, and the existence of

nonl inear rad ia t ive phenomena l i ke anomalous magnetic moment and Lamb shCft even in

c lassical mechanics.

Let us separate in Eq.(1) the se l f f i e l d term:

moZ ~ = e~ FeXt(x) Z~ + eFself(x)pv ~v ~v x=z

(4)

where I have introduced a parameter ~(~=1) in order to study the l i m i t ~ ÷ 0 for a

free pa r t i c l e . The f i r s t term on the r i gh t hand side of (4) is f i n i t e , but the se-

cond term becomes i n f i n i t e at X=Z. By various procedures one can however study the

st ructure of th is term 121. The resu l t is as fo l lows. The s e l f - f i e l d term in (4)

can be wr i t t en , using (3), as a sum of two terms

e 2 - ~÷olim 2--~-Z~ + ~ e2 (Z"" + ~ 2 ) (5)

Here Z~ depends on x as we l l , Zp = Z (S,X)... The f i r s t term is an i n e r t i a l part

which we wr i te as -6m Z and bring i t to the l e f t hand side of (4). We shal l now

"renormalize" eq.(4) such that for x ÷ 0 we have the free par t i c le eq. m~ Z = O. The

renormalizat ion procedure is not unambiguous: we have to know to what form we want

to br ing our e q u a t i o n s . The above r equ i r emen t f o r x ÷ 0 g ives us the f o l l o w i n g f i n a l

equation

~ = e~ Fext.(x)~v ~v +2~ e 2 (Z"" + ~ 2 ) _L °I'2e2 ('~ ÷ ~ "~2)]] , (6) m R ~ x=O

= + 6 m m R m o

Had we not subtacted the las t term, a "free" pa r t i c l e (~ = O) would be governed by

a complicated equation, and that is not how mass is defined. Also, eq.(6) , shows

without the las t term the pecul iar phenomena of preaccelerat ion and socalled run

away solut ions 131whichhave bothered a l o t of people up to present time. The las t

term in (6) el iminates these problems.

The nonl inear term in (6) has a l l the correct physical and mathematical prope L

t ies :

I ) ? Z~ = O, where r = C (i" ÷ Z 22),

2) I t gives correct rad ia t ion formula and energy balance.

3) I t is a non-perturbative exact resu l t .

I t has moreover, the physical i n te rp re ta t ion as Lamb-shift and anomalous

magnetic moment. These can be seen by considering external Coulomb or magnetic

f i e lds and evaluat ing i t e r a t i v e l y the e f fec t of the rad ia t ion react ion term 141.

The c lassical theory can be extended to par t i c les with spin 151. The spin va-

r iab les are best described today using quant i t ies forming a Grassman algebra 161. The

main resu l t , except for addi t ional terms, is the same type non- l inear behaviour ra-

d ia t ion term as in eq.(6).

Some so lu t ions of the rad ia t i ve equations w i th spin are known 171. They exh i -

b i t much of the t yp ica l e r r a t i c behavior of the t r a j ec to r y around an average t r a j e ~

to ry which we know from the Dirac equation as "z i t terbewegung". Conversely, the

c lass ica l l i m i t of the Dirac equation is not a sp in less p a r t i c l e , but a p a r t i c l e

w i th a c lass i ca l spin. Thus the spin of the e lec t ron must be an essent ia l feature

of the s t ruc tu re of the e lec t ron (not j u s t an inessent ia l add i t i on ) .

I I I . QUANTUM THEORY OF SELF-INTERACTION

We see thus tha t the e lec t ron 's equation of motion is fundamental ly non- l i near .

When we go over to quantum mechanics we do not quant ize the " r ad i a t i ng , s e l f - i n t e -

rac t ing e lec t ron" but f i r s t the free e lec t ron . Let us compare the c lass ica l and

quantum equations p a r a l l e l y :

m Z = e ~ Fext ' (x=Z)Z ~ + e Fsel f (x=Z) ZU o p ~ p~

( - i y p ~p - m)~ = e yUA (x) ¢,(x) + ? , (7)

o r , n o n - r e l a t i v i s t i c a l l y and fo r A = O,

~2 (i~i ~ - ~ A ) ~ = U ~ + ? (8)

We see that in the standard wave mechanics the non l inear terms coming from s e l f -

f i e l d have been omit ted, and a renormalized mass have been used. But th i s is only

an approximation. Hence we wish now to complete the wave equations by the inc luss ion

of the s e l f - f i e l d terms.

Nonl inear terms have been added to (7) and (8) in order to have s o l i t o n - l i k e

so lu t ions 181, 191. I should l i ke to discuss here the non- l i near terms in the stan-

dard theory, thus w i thou t in t roduc ing any new parameters.

We consider the basic framework of Lorentz, eqs . (1 ) - ( 3 ) , but when the e lec t ron

is described by a Dirac f i e l d ~(x) :

( - i y P ~ - mo) ~ : eyP~(x)A (x) (9)

F v '~ (x ) = j (x) = e~(x)y ~(x) . (10)

In the gauge A u = O, we can e l iminate A (x) from these equations and obtain the

non- l i near i n t e g r o d i f f e r e n t i a l equations

( - i y u ~ - m o) @(x) = e2~U@(x) ]dyD(x-y)@(y)¥~@(y) .

The choice of the Green's function is not unique. Knowing the par t ic le interpreta-

t ion of the negative energy states in the hole theory, we may choose the causal

Green's function D.

Again, as in the classical case, th is equation is not yet complete: we must

make sure that in the absence of external interact ions the par t ic le obeys the equa

t ion

( - i y ~ - mR)@(x) = 0 (11)

For more than one par t i c le we may e i ther introduce several f ie lds 41 , 42 . . . . .

or consider nonoverlapping local ized solutions of one f i e l d . In the former case,

we obtain the coupled set of equations:

( - i y ~ _ m~l))@l(X) = ely2 p~l, Ix~' i dy D(x-y) ~l(y) ~@1(y)

+ e le2Y~l(X) IdyD(x-Y)~2(Y)yU@2(Y) + (12)

and s im i la r equations for 42, 43 . . . . Here the f i r s t term is the se l f - i n te rac t ion ,

the others interact ions of other par t ic les . For a local ized 42, for example, the

second term in the stat ionary state gives correct ly the interact ion potential as

r ° ' " , r

so that the Coulomb potential is modified at small distances, as i t should be. Now

l e t us wr i te eq.(12) as

_ _ y u ~ . e x t . • , (-i~P~p m o ~p )@~x) = e2yP@(x) Idy D(x-y)~(y)y~@(y) , (13)

where again ~ is a parameter of the external potent ia l . Let ~ ( x ) be a solut ion o f

the l e f t hand side. In the i t e ra t i ve solut ion of the f u l l equation, the r igh t hand

side gives, i f we inser t the solut ion of the homogeneous equation ~ ( x ) ,

l~(x) ~ e2~U~(x) Idy D(x-y) ~ ( y ) yule(y) (14)

For ~ = O, we must have the free eq.(11). Hence l~:o(X) must be used to renormalize

the mass, and perhaps the f i e l d @(x).

I f we work with localized functions always, the theory, including renormaliza tion procedure, is f in i te , and nonperturbative; i t describes a dressed, radiating self-interacting particle.

We consider now in a bit detail the two-body coupled equations for ~ and n:

with

(y~P ml)~ = e yUA (1)self e UA (2) - i ~ ~ + i Y

(~UPu - m2)q = e2~A(2)selfu n + e2~AJl)

A(1)(x)u = e I Idy D(x-y) ~(y)yu~(y)

A(2)(x)u = e 2 idy D(x-y) ~(y)~ q(y)

We shall bring these equations into a manageable radial form using the ansatz

r ( i ) t -iL m #(i)(x ) = ~ #~i)(~)e n

where n labels the quantum numbers (En,J,M,K) and

l i ( i ) ( r ) ~ i ) ( ~ i ~ gn , ~JiM (~) ~ ~C JM Y~(~) ×~ ,

Im;½

with these substitutions, and

d4k e -ikx D(x) = - ~ k 2+i~

(15)

(16)

(17)

(18)

one obtains after much computations the coupled radial equations 1101

2 df s Ks-1 l e l ~ - [

= dr' dr r fs + (Es-ml)gs 2 - ~ 7 Es=Em-En+Er j VlEnEm

n.m.l

I [ ~'m'sr] [g'n* f ' Tnm' sr' ' T nmsr + f ' * f ' T + f r x gr gn* gm-1 n m m-2

s> f { i ele2 [ - , * ,~nmsr + dr' ( r . r ' ) gr V1EnE Len eml 1 27 r 2 E m En+E r m

Tnm'sr' _ dn. e' n'msr'] 2 m T2 [e~* d' + fr m

( r , r ' )

f'*n gm' T~ 'msr']

+ H,*H,T n'm'sr] -n -m-1 ]

K + 1 2 r dgs + s 1 el > r d--r- r gs - (Es + m) fs = - 27 r 2 ]dr ' VIEnE

Es=Em-En+E r m ( r , r ' )

I [ n msr] [ I fr gn* gm-1 + f'*n f'm T1 - gr gn'* f'm -2Tnm's'r-fn*gm' Tn'ms2

1 ele2 > i { ~-e,.e,Tnms'r'+d,.d,Tn'm's'r' ] 27 r 2 dr' V 1 E E ( r , r ' ) f r [ n m 1 n m 1 Es=Em-En+Er n m

- gr L [e~* d' mnm's'r d '* e' Tn'ms'r] m'2 n m'2 ] '

f ' = f ( r ' ) , g' = ( r ' ) , etc. (.19)

There are two similar equations for e r and d r .

Here the Kernels V are known integrals

f= k2dk j~ (k r ) j~ (kr ' ) ( r , r ' ) = - 17 r2r'2 ]

V~EnEm o (En-Em)2- k2 + i

1 En ÷ Em 1 i 3/2 r< ~+~ , T2~+---- ~ (r<r>) (~)

(12o)

T 1 and T 2 are known functions of Clebsch-Gordon coeff icients.

The terms on the r i gh t hand side are the various in terac t ion and rad ia t ive po-

t en t i a l s . To see these more c lear ly we special ize to the s tat ionary Is -s ta te (of

positronium, for example): I = 0 , K = - I , a l l J = 1/2, etc. Then

df + 2 1 I I VOEIE1 d-r ~ f + (El - ml) g = 2~ r 2 g dr' ( r , r ' )

- I d r ' ( r , r ' ) x e (g 'g ' * + f ' * f ' ) + ele 2 (e '*e ' + d ' *d ' ) ~-~ ~- f

× [e21 f ' g ' - e l e 2 d ' e ' ] (21)

and s im i l a r l y the other equations. We shall refer to the f i r s t term as " e l e c t r i c " ,

to the second term as "magnetic" po ten t ia l , because they are mul t ip l ied by g and

f , respect ively.

One can see by e x p l i c i t ca lcu lat ion that the contr ibut ion of the second par-

t i c l e , also in s-state, to the f i r s t gives an e lec t r i c po ten t ia l , in the l i m i t

r ÷ ~ ,

ele 2 - - g + . . . . ( 2 2 ) 4~r

as we have noted ea r l i e r , and as r ÷ 0

Z~m Y4---~g = const, g. (22')

The magnetic potent ia l behaves l i ke

1 2 y + l ele2 f 8m 3 r 2

1 (Z~) 3 m 2 - 3 4~y(2¥- I ) r f

, as r ÷ ~

, as r ÷ O

(23)

I f the second par t i c le is heavy for example, we can use the Coulomb potent ia l only

and obtain

10

df d r

K-I r

2 e eI >r f + (E-m - A~--~) g = r2 g VOE E ( r , r ' )

4K2 ~ 1 (g'*g' + f ' * f , ) _ ~ f I od r 'V iEE ( r , r ' ) f ' g '

e~ I [F ( r ) f + G ( r ) g ] - Vmf + - 27 ~ Veg (24)

s im i l a r l y for the other equations.

Here we introduced the e lec t r i c and magnetic form factors G(r) and F(r ) , res-

pect ive ly . The magnetic form factor F(r) has the form (which we shall need la te r )

i ~ - 2 r / r 0

F(r) = C dr' VIEE(r,r ' ) f ( r ' ) g ( r ' ) = C ' ( l - e ( l + p o l y n ( 2 r / r o ) ) , (25) 0

thus star ts from zero and approaches a constant for large distances, a behavior

which we know from perturbation theory.

Let us compare this resu l t with the Dirac equations for the electron with an

anomalous magnetic moment a in the Coulomb f i e l d

ele 2 ele 2 df K-1 f + (E - m - ) g = a f (26) dr T ~ 2mr 2

which is of course va l id for r ÷ ~, hence the i den t i f i ca t i on of the anomalous mag-

net ic moment in teract ion.

The anomalous magnetic moment has also an interact ion with the s e l f - f i e l d . S~

m i l a r l y , we have addi t iona l e lec t r i c potent ia ls , and, as we see from (21), a

charge renormalization due to the term (e /4~) ~ g. But before using these values,

we must renormalize the self-energy ef fects. In fac t , from the integrals evaluated

with the t r i a l Coulomb type functions, for example we must subtract the i r values

when e 2 ÷ O, the free par t ic le values.

Other Remarkable Solutions of Nonlinear Equations

The Dirac equation in Coulomb f i e l d without the rad iat ive terms on the r igh t

hand side, has the well-known discrete spectrum and the continium, the complete

set of solutions is known. We get a h int for a new class of solutions with radia-

t i ve terms corresponding to sharp resonances from eq.(24). Eliminating one of the

rad ia l funct ions f or g we can obta in a Schrodinger type eigenvalue equation 111[

~,, = (m 2 _ E 2 + Vef f (E , r )~ , (27)

where

Vef f = K (K+ I ) + 2E U - U 2 + V 2 + 2V m K V'

r 2 m ~ - m

U" - 2U' K 'V I ~ - 2U m + 3 U '2

U)2 ' + 2 m+E-U 4 (m+E-

e le 2 U = V I +4-Tr -

I f we look at the shape of th is po ten t ia l fo r V 1 = O, fo r exampl~we see that

beside the usual Coulomb and cen t r i fuga l b a r r i e r , we have a ra ther large po ten t ia l

wel l at small distances and then the po ten t ia l goes to +~ as r ÷ O. For po ten t ia l s

of th is type very high energy narrow resonances have been located numer ica l ly 1121.

There is a simple i n t u i t i v e p ic ture fo r these resonances and they can be even calcu

la ted in semi-c lass ica l r e l a t i v i s t i c theory: They are the unstable states bound due

to magnetic forces. They can be ca l led the magnetic leve ls in contrast to the e lec-

t r i c leve ls o f the Coulomb problem. The corresponding two-body states in the case

of (e + - e - ) have been ca l led superpositronium. They may be i d e n t i f i e d with the high

energy narrow ~-resonances in the (e+e -) system 1131. We also p red ic t s i m i l a r re-

sonances fo r (e-e-) system.

Going back now to our non l inear in tegra l equation (24), we assume a l oca l i zed

so lu t ion of some size r o, evaluate the form fac to r F ( r , ro ) wi th th is form fac to r we

p lo t Vef f in (27) and look for a resonance with pos i t i ve energy E = Ere s. The form

fac to r changes the form of the po ten t ia l only at distances about r ~ r o. The impor-

tant s ize determining the pos i t i on o f the po ten t ia l wel l is given by the value of

the anomalous magnetic moment, ioe. ~2/m. In p r i n c i p l e one can i t e r a t e and look fo r

a se l f - cons i s ten t so lu t i on : the wave funct ion of the resonance must match f a i r l y

wel l the i n i t i a l t r i a l func t ion .

Some Related Problems

(1) The se l f -energy theory can be extended from e lec t ron to neut r ino , in p a r t i -

cu la r , the l i m i t e ÷ O, m + 0 such that e/m = const, impl ies an anomalous magnetic

moment fo r neut r ino. The problem of anomalous magnetic moment of ~ has a long his-

tory . Here we wish to po in t out that the mechanism of magnetic resonances that we

have discussed could also occur fo r the system (ev) or (uv), or even perhaps fo r

( ~ ) . Furthermore, how much of the weak in te rac t i ons can be simulated by the anoma

12

lous magnetic moment in teract ions is an open problem. The magnetic resonance pheno

mena can fu r ther be extended to three (or more) body systems, l i ke (eee), (ee l ) ,

(ee~),etc.

(2) The sel f-energy in te rac t ion of the electron may give r ise to a kind of

"exci ted" state of the electron in the form of ~. A heur i s t i c ca lcu la t ion of

m/m e = + 1, on th is basis has been given elsewhere 1141.

(3) As we have noted, the in terva l s t ructure of the electron is essential for

a closed complete and consistent theory of the electron and electrodynamics. This

is also essential for the understanding of quantum p r inc ip le . A ca lcu la t ion of the

Planck's constant ~ in the quantum rad ia t ion energy E = ~ , using the st ructure of

the electron (of the type given by "z i t te r -bewegung") and the rad ia t ion formula

1151 may i l l u s t r a t e s fu r ther the importance of nonl inear e f fects .

(4) I t would be very in te res t ing to f ind the classical r e l a t i v i s t i c l i m i t of

our s e l f - f i e l d in tegra ls (13) by the method of Rubinow and Kel ler 1161 to see i f

the Lorentz-Dirac Eq.(6) is reproduced.

13

REFERENCES

4

5

6

7

8

9

110 tl1 112 113

114

115 116

P.A.M. Dirac, Proc. High Energy Physics Conf. Budapest 1977.

P.A.M. Dirac, Proc. Roy. Soc. (London) A167, 148 (1938). A.O. Barut, Phys. Rev. 1__0_0, 3335 (1974), and references therein.

G.N. Plass, Rev. Mod. Phys. 33, 37 (1961). F. Rohrlich, Classical Charge-d-Particles, Addison-Wesley, Reading, 1965.

A.O. Barut, Phys. Lett. 73B, 310 (1978).

H.J. Bhabha and H.G. Corben, Proc. Roy. Soc. (London) A178, 273 (1941).

F.A. Berezin and M.S. Marinov, Ann. of Phys. 4, 336 (1977).

H.C. Corben, Phys. Rev. 121, 1833 (1961), and Nuovo Cim. 2_0_0, 529 (1961).

A.F. Ramada, Intern. J. Phys. 16, 795 (1978).

I. Bialynicki-Birula, J. Michielsky, Ann. Phys. 100, 62 (1976).

A.O. Barut and J. Kraus, to be published, and Phys. Rev. 16___DD, 161 (1977).

A.O. Barut and J. Kraus, J. Math. Phys. 1__7_7, 504 (1976).

A.O. Barut and R. Raczka, Acta Physica Polon. (in press).

A.O. Barut and J. Kraus, Phys. Lett. 59B, 175-178 (1975).

See Ref. 141.

A.O. Barut, Z.f. Naturf. 33a, 993 (1978).

S.I. Rubinow and J.B. Keller, Phys. Rev. 131, 2789 (1963).

ON THE STABILITY OF SOLITONS

I. Bialynicki-Birula

Inst i tute of Theoretical Physics, Warsaw University and De-

partment of Physics, University of Pittsburgh, Pittsburgh.

Table of contents

Page

I. Liapunov Stab i l i t y 16

I I . Poincar6 Stab i l i t y 17

I I I . Perfect Poincar~ Stab i l i t y 18

IV. Logarithmic Schr~dinger Equation 19

V. Relat iv is t ic Systems 20

VI. Gauge Fields 23

References 27

16

ON THE STABILITY OF SOLITONS

I . B ia l yn i ck i -B i ru la

I ns t i t u t e of Theoretical Physics, Warsaw Univers i ty and De-

partment of Physics, Univers i ty of Pi t tsburgh, Pit tsburgh.

I . LIAPUNOV STABILITY

The notion of s t a b i l i t y has several meanings both in mathematics and in phy-

s ics. In the studies of so l i tons one usual ly uses the notion of s t a b i l i t y derived

from the notion of stable equ i l ib r ium in c lassical mechanics. A mechanical system

is in a state of stable equ i l ib r ium i f the energy of the system at ta ins i t s minimum

(at least l o c a l l y ) . Small osc i l l a t i ons around the state of stable equ i l ib r ium can

be decomposed in to a superposit ion of harmonic modes of osc i l l a t i ons . The Hamilto-

nian describing these osc i l l a t i ons is obtained by re ta in ing only quadratic terms

in the expansion of the f u l l Hamiltonian around the point of stable equi l ibr ium.

The point of stable equ i l ib r ium, therefore, is characterized by two condi t ions:

~E = 0 (1)

~2E = 0 (2)

Small osc i l l a t i ons around the point of stable equ i l ib r ium are stable in the

sense of Liapunov. Alexander Liapunov, a Russian mathematician, introduced the no-

t ion of s t a b i l i t y in the theory of d i f f e r e n t i a l equations. The Liapunov s t a b i l i t y

of a so lut ion of a d i f f e r e n t i a l equation, or a set of d i f f e r e n t i a l equations, can

be defined whenever the i n i t i a l value problem is well posed. A solut ion Go(t) is

cal led stable i f small perturbat ions of the i n i t i a l data at t =0 lead to small

changes of the so lu t ion fo r a l l t>O.

For par t ia l d i f f e r e n t i a l equations describing so l i tons ( th is term is used here

to denote a l l local ized solut ions with f i n i t e energy) the notion of Liapunov stabi

l i t y must be modified. In most cases of in te res t we are dealing with equ i l ib r ium

which is not stable but neutral with respect to some perturbat ions. Such perturba-

t ions represent "free motions" and are connected with the symmetries ( t rans la t i ona l ,

r o ta t i ona l , d i l a t a t i o n a l , e tc . ) of the system. Only those perturbat ions which do

not represent "free motions" sa t i s fy the Liapunov s t a b i l i t y condi t ion. The discu-

ssion of s t a b i l i t y along these l ines is given in every thorough review a r t i c l e on

so l i tons (see for example I l I ) . In my lectures I shal l introduce and explore a d i -

f fe ren t notion of s t a b i l i t y , which can be traced back to Henri Poincar~.

17

I I . POINCARE STABILITY

At the beginning of th is century Poincar~ has been working on the s t ruc ture of

the e lec t ron. He was the f i r s t to discover that i t is d i f f i c u l t to reconci le r e l a t i

v i s t i c invar iance with the assumption that the dynamics of the e lect ron is governed

by purely electromagnet ic forces. In a somewhat more modern language we can express

his problem as fo l lows. The fundamental physical quant i ty descr ib i rg an extended

system in a r e l a t i v i s t i c theory is the energy-momentum tensor T , whose components pv

are in te rp re ted as the energy densi ty (Too), the energy f l ux densi ty (To i ) , or the

momentum densi ty , and the stress tensor ( T i j ) . For the electromagnet ic f i e l d these

components are:

1 ~2) (3a) To O = 2 (~2 +

Toi = (E x B ) i (3b)

1 (~2 + ~2) T i j = - EiEj - BiBj + 6 i j T (3c)

Let us consider the simplest case of a s ing le , spher i ca l l y symmetric, extended

p a r t i c l e . In i t s res t frame, in view of the symmetry, Toi = O. Under a special Lo-

rentz t ransformat ion (a boost in the x d i rec t ion ) the components of T transform

in the fo l lowing manner:

Too + B2TII 'T = (4a)

oo 1 - B 2

+ Too T l l

'T = 01 S (4b) 1 - 62

where B = v/c and nonprimed components re fe r to the res t frame. In tegra t ing Eqs.(4)

over ' x , 'y and'z, we obtain

I / d3Jx,T _ 1 d3x (Too + 132Tll) O0 ~ o

(5a)

f f _ 1

j d3 'X 'To l 1 _ ~ Jd3x (Too+T11) (5b)

where we used the re la t i on d31x = d3x ~ 1 - 92 . The formulas (5) agree with r e l a t i -

v i s t i c t ransformations laws for the energy E = Id3x Too and momentum P1 = I d3x To1:

'E : El/1 - B 2 , PI = El/1 - B 2 (6)

18

only when the integral of T 11 we can derive the same conditions for T 22

vanishes. Changing the di rect ion of the Lorentz boost,

and T 33'

Id3x T i i = 0 ( i = 1,2,3) (7)

For purely electromagnetic forces conditions (7) can not be sa t i s f ied , because

T11 + T22 + T33 = ½ (~2 + ~2) > 0 (8)

In order to circumvent this d i f f i c u l t y , Poincar~ suggested in his paper on the dy-

namics of the electron 121, that one should introduce addit ional forces of a non-

electromagnetic o r ig in , the so cal led Poincar~ pressures, acting at the surface of

the electron. These forces should be so chosen that they give contributions to the

stress tensor with the opposite sign and make a l l integrals of the diagonal compo-

nents of T i j equal to zero. Poincar~'s cohesive pressure compensates the explosive

electromagnetic force and gives the electron a certain degree of s t a b i l i t y . Howe-

ver, the s t a b i l i t y is not perfect, because only the integrals of the stress tensor

vanish; the stresses are balanced merely on the average. I shall cal l Eq.(7) the

Poincar~ s t a b i l i t y condition. Every r e l a t i v i s t i c local ized system with f i n i t e ener-

gy sa t i s f ies these conditions. An in terest ing and elegant example of such a system

is the model of the electron described by the nonlinear electrodynamics of Born

and Infe ld 131.

I I I . PERFECT POINCARE STABILITY

The Poincar~ s t a b i l i t y condition (7) i s , of course, sa t is f ied i f the system is

t o t a l l y s t ress- f ree, i .e . when a l l components of the stress tensor i den t i ca l l y va-

nish

T.. = 0 (9) i j

I shal l cal l Eq.(9) the perfect Poincar~ s t a b i l i t y (PPS) conditions. This strong

version of the Poincar~ condition is rooted in the general theory of r e l a t i v i t y , s~

m i la r l y as the weak version (7) resulted from the special theory. Indeed the Eins-

tein f i e l d equations, which form the mathematical basis of th is theory, can be v ie-

wed as a fourdimensional version of the PPS condition (9). This is emphasized below

by the appropriate choice of notation:

where

T t ° ta l = T matter + T space-time = 0 (10)

Tspace-time c 4 ~ = 8xk ( R - ½ g ~ R) (11)

19

Thus, gravitating matter is perfectly stable since al l i ts stresses are counterba-

lanced by the stresses induced in the space-time continuum. The r ig id i t y of space-

time is enormous; in the CGS units the coefficient c4/8xk is of the order of 1048 .

For example, to counterbalance the stress due to the electr ic f ie ld , the space need

only bend I part per 1027 for every V/cm. In what follows I shall explore the sign~

ficance of the PPS condition outside the scope of the theory of gravitation. Even

though this condition is primarily meant to apply to re la t i v i s t i c theories, i t also

can be used without re la t i v i t y .

IV. LOGARITHMIC SCHRODINGER EQUATION

We have come across the PPS condition for the f i r s t time in our study with Jer

zy Mycielski 141 of nonlinear wave mechanics. Searching for a nonlinear, Schrgdin-

ger-type equation, whose solutions could be viewed as bona fide wave functions, we

found the equation with the logarithmic nonlinearity.

i~ ~t ~ ( r , t ) = N A - b ln( l~12a n) ¢(~, t ) (12)

where a and b are constants and n is the dimensionality of the configurat ion space.

Our equation is a member of the fol lowing class of nonlinear equations:

i~ a t , ( r , t ) = N A + U(~) + F(I,I 2) ¢(~,t) (13)

The stress tensor for any equation of type (13), with a real function F, can be ob-

tained from the hydrodynamical form of the nonlinear Schr6dinger equation 141:

To . = 1j

where

and

@t p + v. j = 0

m at j + v.T + mpv U = 0

: i~12

")" 2 m ~ ~K--P j = ~ v ¢

~i 2 (@i~j~*.@+~*.ai~j~-~i~*.aj~-@j~*.~i ~) + ~ijP [F(p) - G(p)]

(14a)

(14b)

(15a)

(15b)

(15c)

1 ~d ' F(p') (16) G(p) = T P

^ The current j and the stress tensor T can also be expressed in terms of density and velocity:

20

J = P ~ (17a)

Tij - p L4 m 3i3 j In p + 6i j (G - F) + mpviv j (17b)

In the soliton rest frame the velocity ~ is zero and the PPS condition takes on the

form: ~i 2 4~ 3i~j In p + 6ij (G - F) = 0 (18)

One can show (cf. 14I) that the only solution of this set of equations is the loga-

rithm for F and the Gaussian for p:

-2mb I ~ - ~I2/~i 2 F = - b In (pA) , p = N e

where A,b and N are a rb i t ra ry constants.

In th is way we arr ive at the Schrodinger equation with the logar i thmic non l i -

near i ty . This is a nonl inear equation which in any number of dimensions possesses

so l i ton- type solut ions (we ca l l them gaussons) obeying the PPS condi t ion.

V. RELATIVISTIC SYSTEMS

Now I shal l turn to r e l a t i v i s t i c f i e l d theories. F i r s t , I would l i ke to point

out that here again one dimensional f i e l d theories are except ional ; every local ized

and s ta t i c so lu t ion of r e l a t i v i s t i c f i e l d equations obeys PPS. This fol lows d i r ec t l y

from the con t inu i t y equation for the energy-momentum tensor. In one space dimension

the space component of th is equation reads:

3 o Tol + 31 T l l = 0 (19)

For static solutions 3oTol = 0 and therefore 31Tll = O. For localized solutions this

implies T l l = O.

In more than one dimension the conditions T.. = 0 impose, in general, very i j stringent restrictions and for simple theories they just can not be satisfied. One

example has already been mentioned in these lectures: i t is the pure electromagne-

t ic theory. The stress tensor can not vanish there, because i ts trace is positive.

Before I give examples of f ie ld theories in which the PPS condition can be satis-

f ied, le t me make a few remarks on the energy-momentum tensor.

In order to derive an expression for this tensor in a given re la t i v i s t i c theo-

ry we can use the standard prescription of the general theory of re la t i v i t y , even

21

though g r a v i t a t i o n w i l l p lay no ro le in our d iscussion. According to th i s p rescr ip -

t i on a l l ten components o f the energy-momentum tensor are obtained as de r i va t i ves

of the ac t ion w i th respect to the ten components of the metr ic tensor gPV,

= 2 ~ Id4x C-g L T (x) ~gPV(x) J

(20)

where g is the determinant of the matr ix guy" The energy-momentum tensor def ined by

t h i s formula, as a r esu l t of f i e l d equations w i l l s a t i s f y the c o n t i n u i t y equat ion,

i f the act ion is i n v a r i a n t under general coordinate t ransformat ions 151. For exam-

p le , f o r the Maxwell f i e l d we obta in :

(21) L = - ¼ fi~x fvp guy g~.p

where we have used the fo l l ow ing property of the determinant:

Formulas (20)-(22) lead to

T

6g = - g guy 6gpv (23)

f~. 1 fx p = f~. v + , guy f~.p (24)

The de r i va t i on of the energy-momentum tensor fo r a s e l f - i n t e r a c t i n g sca lar f i e l d

is equa l l y s t ra igh t fo rward :

L = a @*'~v @ gpv - V(@*@) (25)

= * ~* ~ Tpv ~ ~ ~ ~ + ~u * - ~ ~#~ g ~ (~X ~ V(~* ~ ) ) (26)

Here again one can check tha t the PPS cond i t ion in three dimensions has no n o n t r i -

v ia l so lu t i ons .

The stress tensor can also be obtained from the Hamiltonian by d i f f e r e n t i a -

t i ng i t w i th respect to space components o f the metr ic tensor. Since the time com-

ponent and a l l mixed components of the metr ic w i l l not be var ied, I sha l l choose

them in t h e i r Cartesian form

goo = 1 , goi = 0 , g i j = - h i j (27)

22

where hi j is a posit ive def ini te,3x3 matrix. In evaluating the stress tensor from

the Hamiltonian one must, however, be careful and before calculating the der ivat i -

ves with respect to g i j one has to perform the Legendre transformation and express

the Hamiltonian in terms of canonical variables. 0nly af ter this transformation the formula

T i j = 2 ~--C--Id3x~h- H(~) (28) ahiJ

can be used. In the theory of the electromagnetic f ie ld the Legendre transformation

takes us from (B,E) variables to (B,D) variables (see for example 161)

D k ~ hl/2 hl/2 h kl = @ T k L = E l (29)

and we obtain for the Hamiltonian:

H = ½id3x h -1/2 (DiD j hij + BiB j hi j)

In the calculation of the stress tensor the following relations are useful:

= _ hikahklhl, j = . h - I 6hij , ~h = - hhi j 6h i j , Sikm Ejln hkl hmn 2hi3

(30)

The resul t of this calculation coincides with the Maxwell stress tensor, when

hi j = 6 i j . In the theory of the charged scalar f ie ld the Legendre transformation

takes us from (4,~) variables to (4,~) variables,

hl/2 hl/2 ~* = ~ L = (31) ~4

and the Hamiltonian is:

H = Id3x v~- (~*~h - I + ~i4*~j4hiJ + V(~*4) ) (32)

The stress tensor obtained by taking the derivatives of (32) with respect to h IJ,

in Cartesian coordinates in f l a t space is:

T i j = Bi 4" @j4 + ~j 4" Bi 4 + 6i j ( 7 * x - v 4 * . v 4 - V) (33)

This expression coincides with the space part of (26).

The expression (28) for the stress tensor enables us to interpret the PPS con-

d i t ion as the condition that the energy is stat ionary with respect to al l perturba-

23

tions of the space metric h i j . Does one also obtain the true minimum of the energy?

As we shal l see la te r , for several in terest ing systems, i t is indeed so.

VI. GAUGE FIELDS

By comparing formulas (3c) and (33) we in fer that i t is advantageous from the

point of view of PPS to combine scalar f ie lds and vector f i e l ds , because these two

f ie lds contribute to the stress tensor with opposite signs and a cancellat ion may

take place. In two space dimensions such a cancellation may take place even with

one vector f i e l d , i . e . with the electromagnetic f i e l d 171 . However, in three dimen-

sions we need more than one vector f i e l d . Systems of vector f ie lds (Yang-~lills

f i e lds ) have been extensively studied in recent years and the i r coupling to scalar

f ie lds (Higgs f ie lds ) has also been thoroughly investigated. The ground i s , there-

fore, well prepared to undertake the study of the PPS condition in the context of

those gauge theories. I shall choose the best known example of such a theory: the

SU(2) symmetric Yang-Mills-Higgs theory. There are three f ie lds of each type in

th is model of the gauge theory and the Lagrangian with no se l f - in terac t ions of the

scalar f ie lds is :

L = - ¼ F a F apv + ½ (v~)a(vP~) a

F a = ~ A a ~ A a + A b A c ~ V ~ V l V ~ Cabc ~ v

A b @c (v @)a = ~ @a + Sabc

(34)

(35)

(36)

where the indices a,b,c take on values 1,2 and 3. The stress tensor can be obtai-

ned e i ther by varying the action with respect to guy or by varying the Hamiltonian

with respect to h l j , The second method leads to more interest ing resul ts. The Ha-

mi l tonian, wr i t ten in terms of canonical variables is :

H= ½ Id3xhl/2{DaDahiji j +21- Baij Baklhikhjl + ~a~ah-1 + (vi@)a(Vj@)a hi j }

where

(37)

D i = h 1/2 h i j F a B a = - F a (38) a oj ' ik ik

a = hl/2 (Vo@)a (39)

In order to study the PPS condition i t is convenient to rewrite the Hamiltonian in

a d i f fe ren t form, fol lowing the approach of Coleman et a l . 181, who investigated the

lower energy bound in f l a t space.

24

H = } Id3x h l / 2 { [(D~ h - I / 2 - s i n ~ h i j (v j@)a) (D~ h - I / 2 - sin m h kl (Vl#)a) +

+ (B h - I /2 h - I /2 cos h kl - cos~h i J ( v j@)a ) _ (Vl@)a ~ hik + a a h - l } +

+ sin ~ ; d3x Di I a (vi @)a + cos ~ d3x B i a (vi @)a (40)

where B i , } i j k B a a = jk " The f i r s t integrand in th is formula is a sum of. . three posi-

t i ve quadrat ic forms. Therefore, the minimum of H as a funct iona l of h l j is ob ta i -

ned when each term under the f i r s t in tegra l vanishes, i . e . when ( in f l a t space)

= ÷ ~ = ÷ a = 0 (41) a sin ~ (v~) a , a cos ~ (v~) a ,

Eqs. (41) also guarantee that t h e i r so lu t ion s a t i s f i e s the PPS cond i t ion , because

the las t two terms in the Hamiltonian do not depend on the metr ic tensor h I j . A l l ÷ ÷

we have to check now is tha t Eqs. (41) also imply that D,B,@ and ~ sa t i s f y the

f i e l d equations. This can be ra ther eas i l y done. From Eqs. (41) i t fo l lows that

÷ ÷

( v . v @)a = 0 (42)

÷ ÷ ÷ ÷

since B Is a covar iant cu r l , B a =(vx A)a.E q. (42) is the cor rec t f i e l d equation fo r

the sca lar f i e l d , since ~ = O. The equation fo r

(~.~)a = 0 (43)

fo l lows from Eqs.(41) and (42). I t is somewhat

remaining f i e l d equation is s a t i s f i e d :

To th is end we not ice that

more complicated to check that the

(~×B)a - (Vo D)a = ]a (44)

+ ÷ = = - ~b@c = - (#~) @c (V xB) a cos~ (~ ÷ ~ a b c b xvo) a cos c°s2e ~abc

(Vo~) a = sin ~ (Vo~@) a = s i n ~ a b c ~b @ c = sin2m~abc (V@)b÷ @c

÷ ÷

Ja = - Eabc (V#)b Oc (45)

The so lu t ion of f i e l d equations determined by Eqs. (41) is known in the l i t e r a t u r e

25

as a special case of the Julia-Zee dyon 191. Coleman et al . have shown in 181 that

dyons are stable under al l perturbations of the f i e ld . I have shown above that they

are also stable under al l perturbations of the space metric. As a resu l t of a chan-

ge in the geometry of space, the energy of the dyon can only increase. Following

181 we can minimize the lower bound for the energy by a proper choice of the angle

~, which was so far arb i t rary•

Similar results can be obtained in the space of four dimensions in the world

of instantons. The Lagrangian again has the form given by Eq. (34), but al l Greek

indices now run from 0 to 4. The Hamiltonian has s t i l l the same form (37), but the

number of components of the D f i e l d (four) is now d i f fe ren t from that of the

f i e l d (s ix) . Therefore, we can no longer use the expression (40) to f ind the lower

bound of the energy. Instead, I shall wr i te H, in four space dimensions, in the

forms:

H : ~ Id4x h I /2 [(D~ h -1/2 - h i j (vj#) a (D~ h -1/2 - h kl (Vl#) a kik

+ ¼ (him hJn B a 1 i jmn B a h-1/2) (hkr hlP B a 1 klrp B a mn - 2 mn rp - T ~ rp

• h - 1 / 2 ) h i k hj l + ~a~a] + I d4x Die (vi~)a + ¼/d4x i j k l B elk Bail

The minimum of the energy with respect to h IJ is attained when

D i = (vi~)a , Baij 1 i j k l Ba a = 2 k l ' ~a = 0

(46)

(47)

Again one can check that the solutions of Eqs. (47) are also solutions of the

f i e ld equations. In the simplest case, when ~ = 0 = #, the solut ion is known as

the instanton of Belavin et al. 1101. I t is a self-dual solut ion of pure Yang~Mills

equations in four space dimensions. In four space dimensions, unlike in three, the

trace of the stress tensor of the Yang-Mills f i e l d is i den t i ca l l y zero:

T~ = Fakn Fank + ¼ 6kk Fmn Fmn = 0 (48)

This is why we can have solutions of the pure Yang-Hills equations which sat is fy

the PPS condit ion• For the instanton solut ion we can analyze in fu l l detai l the se-

cond var iat ion of the energy with respect to h l j . For small deviations from f l a t

geometry we obtain

62 H = Id4x T i j , k I 6h i j ah kl (49)

2 8

= B 2 T i j , k l (~ik 6j l + ai l ajk - ~ i j ~kl ) + Baik Bail + Bail Bajk

B2 = ¼ Baik Bika (50)

The 16 x 16 matrix T ik , j I obeys the fol lowing Cayley-Hamilton equation:

12 - 4 B 2 ~ = 0 (51)

from which i t fol lows that the eigenvalues of T are 0 and 4 B 2. Variations of the

metric which belong to the zero eigenvalue do not change the energy. One such va-

r ia t ion is easi ly found: hiJ- = ~6ij-. This is a well know resu l t of the conformal

invariance of the theory. The instanton can be uniformly stretched in a l l d i rec-

tions without a change in i t s energy. Conformal transformations are examples of

" f ree motions" which were mentioned in Sec. I.

In both cases, in 3 and 4 dimensions, we were able to transform the Hamilto-

nian to the form:

H = (non-negative quadratic form) + (metric independent terms)

The h-independent terms depend only on the topology of the solut ions, not on the

Riemannian geometry of the space. They are known (modulo numerical mu l t ip l i ca t i ve

constants) as topological indices, Pontriagin indices or winding numbers. Similar

decompositions of the Hamiltonian can be found in other theories. I do not know of

any general rules that control th is type of behavior.

Analogous resul ts were found independently by Hosoya i l l I, who has studied

the action (instead of the Hamiltonian) for various nonlinear f i e l d theories with

topological conservation laws and discovered the basic decomposition into the qua-

drat ic form plus no-h terms.

27

REFERENCES

1. R. Jackiw, Rev. Mod. Phys. 49, 681 (1977).

2. H. Poincar~, Rediconti Circolo Mat. Palermo 2_j_l, 129 (1906).

3. M. Born and L. Infeld, Proc. Roy. Soc. 147, 552 (1934).

4. I. Bialynicki-Birula and J. Mycielski, Annals of Physics 10___0_0, 62 (1976).

5. A. Trautman, in Gravitation, L. Witten Ed., Wiley, New York, 1962.

6. I. Bialynicki-Birula and Z. Bialynicka-Birula, Quantum Electrodynamics, Perga- mon Press, Oxford, 1975, p. 96.

7. H.J. de Vega and F.A. Shaposnik, Phys. Rev. D14, 1100 (1976).

8. S. Coleman, S. Parke, A. Neveu and C. M. Sommerfield, Phys. Rev. D15, 544 (1977)

9. M.K. Prasad and C.M. Sommerfield, Phys. Rev. Lett. 3_~5, 760 (1975).

I0. A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Yu.S. Tupkin, Phys. Lett. 59B, 85 (1975).

I I . A. Hosoya, Prog. Theor. Phys. 5_.99, 1781 (1978).

SPECTRAL TRANSFORM AND NONLINEAR EVOLUTION EQUATIONS

F. Calogero

I s t i t u t o di Fis ica, Universi ta di Roma, 00185 Roma, I t a l y

I s t i t u t o Nazionale di Fisica Nucleare, Sezione di Roma

ABSTRACT

This is a terse in t roduct ion to the idea of the spectral transform method to

solve nonl~near evolut ion equations.

30

The lecture notes by A. Degasperis printed a f te r th is pape~provide a complete

treatment fo the spectral transform technique to solve nonlinear evolut ion equa-

t ions, based on the approach that uses generalized wronskian re lat ions as the main

tool of analysis 111, and presented in the (rather general) context of the class

of Nonlinear equations solvable via the spectral transform associated to the ma-

t r i x Schr~dinger eigenvalue problem (defined on the whole l i ne , with "potent ia ls"

vanishing at i n f i n i t y ) ; and the subsequent paper by D. Levi surveys the analogous

resul ts for the discret ized case. This "phi losophical" introduct ion is instead con-

f ined to a terse descr ipt ion, in the simplest set t ing, of the idea of the spectral

transform, and is therefore aimed only at those readers whose sole purpose is to

understand the mere essence of th is important mathematical idea.

The standard comparison of the spectral transform technique for solving non-

l inear evolut ion equations is with the Fourier transform method to solve l inear

par t ia l d i f f e ren t i a l equations. Let us reca l l , in the simplest set t ing, how this

works. Let

: - i m (- ~x ) u (x , t ) (1) u t ( x , t )

be a l inear evolut ion equation, m(z) being an ent i re function (say, a polynomial).

The ("Cauchy") problem of in terest is to compute u(x , t ) for t > t o , given

u(x, t o) : Uo(X) (2)

We assume hereafter that al l functions are defined over the whole real axis,

- ~ < x < + = , and that they vanish ( " su f f i c i en t l y fas t " ) as x ÷ ± = ; then the problem

(1-2) is solved by the 3 formulae

I +° a(k, to) : _ dx exp(- ikx) u (x , t o) (3)

Q(k,t) : a(k, to) exp [ - i m ( k ) ( t - t o ) ] (4)

u(x , t ) = (2x) -1 I dk exp(ikx) ~(k, t ) (5) J

Note that the app l i cab i l i t y of th is technique depends on the poss ib i l i t y to go b i -

j e c t i ve l y from configurat ion space to Fourier space (via the inverse Fourier trans-

form (3) and from Fourier space to conf igurat ion space (via the d i rec t Fourier

transform (5)) , and on the fact that to the (complicated) time evolut ion ( I ) in con

f igurat ion space there corresponds the (extremely simple) time evolut ion (4) in

Fourier space. The relevance of these resul ts to mathematical physics is of course

enormous, because many physical (or, for that matter, natural) phenomena are, in

31

some approximation, described by equations of type (2) (or their obvious generali-

zations); indeed i t is here that originates the central rSle played by Fourier ana

lysis in mathematical physics, or more generally in applied mathematics.

The spectral transform technique can be viewed as an analogous tool to solve

and investigate (certain classes of) nonlinear evolution equations. Indeed the me-

chanism is very similar to that described above, but with one main difference; the

mapping between configuration space and "spectral" space (the analog of Fourier

space) is now npnlinear (although i t is generally defined by linear equations).

Let us i l lustrate the situation by a simple, and very recent, example 121.

Consider the nonlinear partial differential equation

ut+ (12t) - I [Uxx x - 6UxU - 4xu x - 2u I : o , u u(x,t) (6)

whose relevance is highlighted by its relation, via the simple change of dependent

and independent variables

u(x,t) = (12t) 2/3 q(y,t) , y = (12t)1/3x (7)

to the so-called "cylindrical KdV equation"

qt + - 6qyq + (2t)-lq = 0 q ~ q(y,t) (8) qyyy

that plays some r~le in plasma physics. Again one is interested in solving the

("Cauchy") problem of finding u(x,t) for t > t o given

u(x,t o) : Uo(X) (g)

(we assume t o > 0, to avoid any problem associated with the singularity of the coef-

f ic ient of the second term in (6) for t =0).

Now associate to u(x,t) a function f (z , t ) through the linear ("spectral") pro-

blem characterized by the second order ordinary differential equation

+ ~x + u(x , t ) ] ~ = z~ , ~ ~ ~(x ,z , t ) (10) ~XX

le t t ing @(x,z,t) and F(x,z , t ) be the solutions of this equation ident i f ied by the

boundary conditions

lim {@(x ,z , t ) /A i ( x - z } = 1 (11a) X~+~

32

F l l im { F ( x , z , t ) / I B i ( x - z ) - i A i ( x - z ) | } : 1 ( l l b )

L J

( that are c lear ly consistent with the assumed asymptotic vanishing of u (x , t ) since

Ai and Bi are the Airy functions 131, so that A i (x-z) and Bi(x-z) are two indepen-

dent solut ions of (10) with u (x , t ) z 0), and def ining

f (z , t ) : [Fx(X,z,t) - F(x,z,t) %(x,z, t ) ] (12)

The correspondence between the function u (x , t ) , - ~ < x < + ~, and i ts "spectral

transform" f ( z , t ) , - ~ < z < + ~ , is biunivocal; the equations wr i t ten above define

uniquely f ( z , t ) from u (x , t ) ; while the computation of u (x , t ) from f ( z , t ) can be

effected via the equations + ~

M(x,x',t) : I_dz [|f(z,t)| -2 - 1] Ai(x-z) Ai(x'- z) (13)

K ( x , x ' , t ) + M ( x , x ' , t ) + I - dx" K(x ,x " , t ) M(x " , x ' , t ) = 0 , x' ~ x (14)

Jx

u(x , t ) = - 2 d K ( x , x , t ) / d x (15)

(The f i r s t of these equations defines the kernel M; the second characterizes uni-

quely the function K ( x , x ' , t ) , being a Fredholm integral equation for i ts depen-

dence on the second variable x ' , while the dependence on the variables x and t is

parametric, or ig inat ing from the e x p l i c i t appearance of these variables in (14);

and the th i rd of these equations defines u). Note that the re lat ion between u and

f is not l inear ( in contrast to the re la t ion between u and i ts Fourier transform 0;

see (3) and (5) ) , although the computation of f from u, as well as the computation

of u from f , requires only the solut ion of l inear equations (the "Schr~dinger"

equation ( I0 ) , and the "Gel ' fand-Levitan" equation (14)).

The so l vab i l i t y of the nonlinear evolut ion equation (6) is now implied by the

(highly nont r iv ia l ' . ) fact that , i f u evolves in time according to th is equation,

the corresponding time evolut ion of f is extremely simple, being given by the ex-

p l i c i t formula

f ( z , t ) = f [ z ( t / t o )1/3, t o ] (16)

so that, to evaluate u(x , t ) from u(x , t o) (see (9)) , one f i r s t evaluates f ( z , t o)

via (10-12)), then f ( z , t ) (via (16)), and f i n a l l y u(x , t ) (via (13-15)), thereby re

ducing the solut ion of the Cauchy problem for the nonlinear evolut ion equation (6)

33

to a sequence of l inear steps.

In addition to providing a technique for the solution of the Cauchy problem

for (6), the association to u(x , t ) of i t s spectral transform f ( z , t ) y ie lds other

qua l i ta t i ve and quant i ta t ive properties of the solutions of (6), for which the in-

terested reader is referred to the or ig inal l i t e ra tu re 121.

As i t has been described above, th is technique appears extremely ad hoc. In

fac t , by appropriate choices of the spectral problem that ins t i tu tes the connec-

t ion between a function and i t s spectral transform, several classes of nonlinear

evolution equations can be solved, including many of considerable pract ical in te-

rest ; and many qua l i ta t i ve and quant i ta t ive properties of the solutions of these

equations can be inferred, foremost among them being the appearance of the solitons,

that are generally associated to the discrete part of the spectral transform, when

i t is present ( th is is not the case in the example we have for s imp l i c i t y selec-

ted-al though even in this case something analogous to the sol i tons could s t i l l be

ident i f ied 121). The interested reader w i l l f ind a detai led treatment of such re-

sults in the fol lowing two papers. Here we conclude noting that the search for no-

vel types of spectral transforms, and the iden t i f i ca t i on of the corresponding

classes of solvable evolution equations, constitutes today a very active and most

promising research l ine .

34

REFERENCES

1. Besides the papers referred to in Degasperis'lecture notes, the following re- view papers are now available: F. Calogero, "Nonlinear e~olution equations sol vable by the inverse spectral transform", in Mathematical Problems in TheoretT- cal Physics, edited by G.F. Dell 'Antonio, S.Doplicher and G.Jona-Lasinio, Lec- ture Notes in Physics 8_00, Springer, 1978; F.Calogero and A.Degasperis, "Nonli- near evolution equations solvable by the inverse spectral transform associated to the matrix Schr~dinger equation" in Solitons, edited by R.K.Bullough and P. J. Caudrey, Lecture Notes in Physics, Springer, 1979; A.Degasperis, "Solitons, Boomerons and Trappons", in Nonlinear Evolution Equations solvable by the Spec- tral Transform, Proceedings of a Symposium held at the Accademia dei Lincei in Rome (June 1977), edited by F. Calogero, Research Notes in Mathematics 26, Pit- man, 1978; F. Calogero and A. Degasperis, "The Spectral Transform: a Tool to Solve and Investigate Nonlinear Evolution Equations", in Applied Inverse Pro- blems, edited by P.C. Sabatier, Lecture Notes in Physics 85, Springer, 1978.

2. F. Calogero and A. Degasperis: "Inverse spectral problem for the one-dimensio- nal Schr~dinger equation with an additional l inear potent ia l" , "Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cyl indrical KdV equation", "Conservation laws for a nonlinear evolution equation that includes as a special case the cyl indrical KdV equa- t ion" , Lett. Nuovo Cimento 2__33, 143, 150, 155 (1978).

3. See, for instance: M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, chapter 10.

SPECTRAL TRANSFORM AND SOLVABILITY OF NONLINEAR EVOLUTION

EQUATIONS

A. Degasperis

I s t i t u t o di Fisica - Universita di Roma 00185 Roma-ltaly

I s t i t u t o di Fisica - Universita di Lecce

I s t i t u t o Nazionale di Fisica Nucleare, Sezione di Roma

Table of Contents

1. The Spectral Transform

2. The Basic Formulae and SNEE

3. Solitons

4. Basic Nonlinear Equations and Related Results

References

Figures

Page

36

48

58

71

84

87

36

SPECTRAL TRANSFORM AND SOLVABILITY OF NONLINEAR EVOLUTION EQUATIONS

1. THE SPECTRAL TRANSFORM

Many l i near evolut ion equations can be solved, and the propert ies of t he i r so-

lu t ions can be invest igated, by means of the powerful method of the Fourier trans-

formation. Indeed, today th is method is i nev i tab l y part of the background of any

s c i e n t i s t , from the pure mathematician to the engineer; of course, the reason is

that i t s relevance and importance has gone well far beyond the theory of heat con-

duct ion, i . e . the subject of the or ig ina l Four ier 's t rea t ise . In the las t decade,

a f te r the important paper by GGKM 111, i t has become clear that , for cer ta in

classes of nonl inear evolut ion equations, the resolv ing method based on the Spec-

t ra l Transform (ST) can be considered 121 as an extension of the Fourier analys is ,

to which i t reduces by l i nea r i z i ng the nonl inear equation (with appropiate caut ion,

see below). I t is therefore reasonable to believe that the ST method opens h igh ly

in te res t ing perspectives in appl ied, as well as in pure, mathematics; in any case,

due to the fortunate fact that many important physical phenomena 131 are modeled by

nonl inear par t ia l d i f f e r e n t i a l equations which are "solvable" by the ST technique,

th is method is ce r ta in ly a major breakthrough in mathematical physics.

By d e f i n i t i o n , a nonl inear evolut ion equation is solvable i f i t s corresponding

Cauchy problem can be solved by means of l i near operations. The ST method consists

on associat ing to a Solvable Nonlinear Evolut ion Equation (SNEE) a l i near eigenva-

lue problem with respect to which the spectral transform of the solut ion of the

SNEE is defined for any f ixed value of t i m e - i n a way which w i l l be detai led below.

Then the spectral transform of the so lu t ion , and th is is the crucia l po in t , turns

out to evolve with time according to a l i near equation; of course, th is scheme

works only i f one is able to inver t the ST and, for th is reason, th is method has

been more commonly referred to as the Inverse Spectral Transform (IST) method. I t

is true that the acronym IST has been introduced by AKNS 12i for the Inverse Scatte

r ing Transform because in the i r paper the associated l inear eigenvalue problem was

a wave scat ter ing problem; however our rede f i n i t i on seems more appropiate since

th is method can ce r ta in l y apply also to SNEE's associated to l i near operators

which do not necessarely describe scat ter ing processes 141.

37

In order to f ind the class of SNEE's associated to a given l inear eigenvalue

problem 151, several approaches have been proposed. The f i r s t systematic way to

produce SNEE's is due to P. Lax 161 and is based on the isospectral time evolution

of a l inear operator; in fact , he proved that the l inear Schroedinger operator d 2

dx 2 + q (x , t ) evolves according to an isospectral transformation i f the "poten-

t i a l " q (x , t ) is a solution of the well-known KdV equation (or of any of the so ca-

l led higher KdV equations). A second approach, which is equivalent to the previous

one and is based on the i n t eg rab i l i t y conditions for a pair of par t ia l d i f f e ren t i a l

equations, has been developed by AKNS 121; within th is powerful scheme, they discu~

sed the important class of SNEE's associated to a non se l f -ad jo in t generalization

of the Zakharov-Shabat l inear problem. A th i rd approach, introduced by F. Calogero,

takes advantage of a generalization of the usual wronskian re lat ions for solutions

of a (generalized) Sturm-Liouvi l le problem 171. This las t technique, showing a close

analogy with the Fourier analysis, seems to be more appropiate to present in a uni-

f ied and transparent way the main results of the theory (such as the B~cklund trans-

formations); i t s appl icat ion to the multichannel Schr~dinger problem w i l l be the

subject of these lectures. In the las t few years, other approaches to nonlinear evo

lu t ion equations have been developed within the framework of d i f f e ren t i a l geometry

181, Lie groups 191 and algebraic geometry 1101.

In the fol lowing we w i l l focus upon dynamical systems described at time t by

a multicomponent f i e l d Q(x,t) depending on one space coordinate x. Solvable nonl i -

near f i e l d equations with more than one space variable can also be obtained 1111 by

the method described below; however, th is non t r i v i a l extension w i l l not be discus-

sed here.

Now we complete this prel iminary discussion by reca l l ing the 3 steps which are

performed to solve the Cauchy problem for the much simpler case of a l inear evolu-

t ion equation. Because we are interested in the basic ideas, we focus here only on

the simpler case of a scalar f i e l d , say q ( x , t ) , sat is fy ing the l inear evolution

equatin~

q t ( x , t ) = - iw ( - i ~x , t ) q (x , t ) , q (x , t o) = qo(X) (1.1)

where m(z,t) is ent i re in z and the subscript t denotes par t ia l d i f f e ren t ia t i on

with respect to time. F i rs t one goes over from the x-space to the k-space via the

Fourier transformation

(

~(k , t ) = ~ dx q(x , t ) exp(- ikx) ~ (1.2) )

38

then the e x p l i c i t time evolut ion of the Fourier transform ~(k , t ) is immediately ob-

tained by integrat ing the fol lowing simple evolut ion equation implied by (1.1)

~ t ( k , t ) : - im(k, t ) ~(k , t ) , ~(k, to) = ~o(k) (1.3)

namely,

q (k , t ) : qo(k) exp [ - i I t d t ' m ( k , t ' ) ] t o

(1.4)

F inal ly the solut ion q (x , t ) at time t ~ t o is recovered by inver t ing the Fourier

transformation

q(x , t ) = (2~) -1 I dk ~(k , t ) exp(i kx) (1.5)

Thus, given q(X, to) , one computes q(k, to) by (1.2), then ~(k , t ) by (1.4), then

q (x , t ) by (1.5) . By appropriately def ining a spectral transformation which maps

from the x-space to the k-space, we w i l l show that th is 3-step scheme applies also

to a large class of nonlinear evolut ion equations. Addit ional important resu l ts ,

which have the i r simple counterparts in the l inear case, w i l l be reported in the

last section.

We s tar t with the de f in i t i on of the ST of an NxN matrix Q(x), depending on

the real variable x; to s impl i fy the introduct ion of the basic mathematical tools,

we generally assume this matrix Q(x) to be hermitian

Q(x) = Q+(x) (1.6)

although almost al l results remain va l id , with obvious modif icat ions, even i f th is

condit ion (1.6) does not hold. We emphasize that the extra e f f o r t needed to deal

wi th a matrix-valued function of x, rather than with a scalar f i e l d , is not due to

a mere sake of general i ty , but i t is required in order to cover almost al l in teres-

t ing evolut ion equations discovered so far. In order to define the spectral trans-

form of Q(x), we associate to i t the matrix Schr~dinger operator

H = -d2/dx 2 + q(x) (1.7)

acting on the L 2 vector valued functions of x. The operator (1.7) and i ts proper-

t ies are well-known in potent ial scat ter ing theory under appropriate assumptions on

the potent ial matrix Q(x). The fol lowing de f in i t i on of ST can apply to a l l those ma

t r ix -va lued functions which sat is fy a l l the usual requirements of the theory of

scat ter ing 1121. However, for sake of s imp l ic i t y , and because i t is su f f i c i en t to

39

cover the more in terest ing cases, we assume stronger conditions on Q(x), namely

that Q(x) is f i n i t e valued ( for real x) and that i t vanishes asymptotically expo-

nent ia l l y or faster , i . e . we assume that, for some posit ive

lim [expI Ixll QIxl]:o 11. I X->-+oo

These conditions together with (1.6) garantee that the operator (1.7) is se l f -ad-

j o i n t and, therefore, that i ts spectrum is real.

The continuous part of the spectrum of the operator (1.7) is characterized by

the matrix d i f fe ren t ia l equation

~xx(X,k) = [Q (x ) - k 2] ~(x,k) (1.9)

together with the fol lowing boundary conditions

~(x,k) ~ exp(-i kx) + R(k) exp(i kx) ( l . lOa)

(x,k) x + - ~ > T(k) exp(-i kx) (1.10b)

Note that here ~ is the NxN matrix depending on x and k which is bu i l t out of N

independent vector solut ions, used as i ts columns; the subscript x means part ia l

d i f f e ren t ia t i on with respect to x and, of course, k is real and, by convention, po-

s i t i ve . The asymptotic behaviour ( l . lOa,b) of the solut ion ~ is characterized by

two NxN matrices: the " re f lec t ion coef f i c ien t " R(k) and the "transmission coe f f i -

c ient" T(k); R(k) depends on the posi t ive real variable k and i ts values for nega-

t ive k can be obtained by hermitian conjugation

R(-k) = R+(k) , (1.11)

the va l i d i t y of th is formula being implied by the results given below.

The discrete part of the spectrum consists of a f i n i t e number of negative e i -

genvalues of the operator (1.7); the fact that the number of these eigenvalues is

f i n i t e is a consequence of (1.8) (of course, th is number may also be zero). Each

eigenvalue is characterized by the posi t ive number p~J)' " which enters into the eige~

value equation

~( j ) ,x , i Q p( j )2] ~( j ) xx ~ J = (x) + (x) , j = 1 , 2 , . . H . (1.12)

40

Without any loss of general i ty , we assume the eigenvalues _p,j 2 f ~ to be nondegene-

rate; in fact , the case of degenerate eigenvalues can be easi ly recovered by a sui-

table l im i t ing process (see below). With th is assumption, the (one column) vector

solut ion #(J)(x) of the equation (1.2) sat is fy ing the normalization condit ion

_ dx ~(J)(x) + ~(J)(x) : i , j = 1,2,.. [I (1.13)

uniquely defines, through i ts asymptotic behaviour

~(J)(x) x ÷ + ~ ~ c ( j ) exp(-P(J)x) ' j : 1 ,2, . . [I (1.14)

the vector c ( j ) corresponding to the eigenvalue _p~jj2.( ~ For future convenience, we

notice that for a large class of matrices Q(x), R(k) and T(k) can be ana ly t i ca l l y

continued in the variable k and are indeed meromorphic in the whole complex k-plane

i f the matrix Q(x) vanishes asymptotical ly faster than exponent ia l ly, i . e . i f for

any real number

lim lexp(mx)Q(x)] = 0 (1.15) X+_+oo

In th is case a de f in i te connection exists between the continuous part and the dis-

crete part of the spectrum,in fact the matrix R(k) turns out to have M simple po-

les at the values k ( j ) = ip (j)," " j = 1,2.. M, the corresponding residues being re-

lated to the vectors c ( j ) by the simple formula 1121

lim { [k ip(J) ] R (k ) }= ic(J) c ( j )+ - ( 1 . 1 6 )

k ÷ i p ( j )

where the dyadic notation has been used. In the fo l lowing, i t w i l l be convenient to

characterize the vector c ( j ) corresponding to the j - t h discrete eigenvalue, with

i t s modulus

p(J) = c(J)+c ( j ) , j = 1,2,.. M (1.17)

and i ts d i rect ion to which we associate the corresponding NxN project ion matrix

p(j) = (p ( j ) ) - I c( j ) c ( j)+ , j = 1,2,.. M (I.18)

sat is fy ing the obvious conditions

p( j )2 = p( j ) , t r P(J) = 1 , j = 1,2, . . M. (1.19)

41

In conclusion, we attach to each point of the discrete spectrum ( i f any) a posi t ive

number p(J) (the eigenvalue being _p( j )z) , a posi t ive number p(J) and a NxN matrix

P(J) projecting on a one-dimensional subspace.

The ST of £he matrix valued function Q(x) is defined as the fol lowing col lec-

t ion of quant i t ies

- ® < < p ' J ' , ( ~ p ( J ) , P ( J ) , j = 1 , 2 , . ~ } ST : {R(k), k + ~, .. (1.20)

The rat ionale of this de f in i t ion w i l l be clear in the fol lowing. Thus, the com

putation of the ST of a given matrix Q(x) requires the solut ion of the d i rect ma-

t r i x Schr~dinger problem, namely the determination of the matrix R(k) and the spec-

t ra l parameters p~J),'' p~J)'' and P(J) through Eqs. (1.9), (1.10), (1.12), (1.13),

(1.14), (1.17) and (1.18). Such a computation defines the mapping

Q(x) + {R(k), p(J), p(J), P(J), j = 1,2,.. M}. Two very important differences

between this mapping (i.e. Q(x) + ST of Q(x)) and the familiar Fourier transforma-

tion should be inmediately underlined; f i rs t , the ST of Q(x) depends clearly in a

nonlinear way on Q(x), and this nonlinearity is the essential property of the ST

which makes possible to solve nonlinear evolution equations. The second difference

refers to the natural splitt ing of the ST into two main parts, namely that one co-

rresponding to the continuous spectrum of the Schr~dinger operator (1.7) characte-

rized by the matrix valued function R(k), and the other one corresponding to the

discrete spectrum of the operator (1.7) characterized by the M pairs of positive

numbers ptJ)' ' and pLJ)' ' together with the i r corresponding M hermitian projectors

P(J). These two components of the ST are completely independent from each other and

w i l l play a very d i f fe ren t role in the appl icat ion of the ST method to the inves t i -

gation of the time evolution of physical systems. Indeed, the very existence of the

discrete component of the ST w i l l be ref lected in the existence of the so-called so

l i tons .

The problem of f inding the ST of a given Q(x) w i l l be referred to as the d i -

rect problem. I ts solution requires the integrat ion of the NxN matrix Schr~dinger

equation. I t is worthwhile to reformulate very b r i e f l y the d i rect problem in terms

of the well-known integral equations of scattering theory 1131 . For sake of sim-

p l i c i t y , and also because we want to focus on the analogy between the Fourier trans

form and the ST, we w i l l consider now only the continuous component of the ST, i .e .

the re f lec t ion coef f ic ient R(k). The s tar t ing point is the integral equation sat is -

f ied by the matrix solution of the d i f f e ren t ia l equation (1.9), together with the

boundary condition ( l . lOb)

42

~(x,k) = T(k) exp(-i kx) + ]_ dy g(x-y,k) Q(y) ~(y,k) (1.21)

where g(z,k) is the appropriate Green function

g(z,k) = ( l /k) sin(kz) 8(z) (1.22)

8(z) being the step function, i .e. 8(z) = 1 for z > 0 and g(z) = 0 for z < O. The

asymptotic behaviour (l.lOa) is then recovered from the integral equation (1.21)

with the following identi f ication

T(k) = 1 + (2ik) -1 exp(ikx) Q(x) ~(x,k) (1.23)

I i ldx expI-ikx I Q(x)~(x,k 1 - (1.24) R(k) = ( 2 i k ) -1

These integral representations of the reflection and transmission coefficients are

not quite appropriate to discuss the problem of obtaining approximate expressions

of T and R in terms of Q. In fact, because of the k -1 factor in front of the inte-

grals in (1.23) and (1.24), the f i r s t order approximation (obtained setting

~(x,k) = exp(-ikx)) would be certainly very poor in the long wave-length l imi t . I t

turns out to be more convenient to introduce the following matrix solution of the

Schr~dinger equation (1.9) 1141

~(x,k) ~ ~(x ,k ) IT (k ) ] -1 (1.25)

which satisf ies the integral equation

~(x,k) = exp(-ikx) + g(x-y,k) Q(y) ~(y,k) (1.26)

and also to introduce, correspondingly, the NxN matrix valued function of two va-

riables

~ (k ' ,k) ~ I i ldx exp(-ik'x) Q(x) ~(x,k) (1.27)

which is i t se l f the solution of the following integral equation

43

- o o

(1.28)

Q(k) being the Fourier transform of Q(x) (see Eq.(1.31) . This integral equation is

derived from Eqs.(1.27) and (1.26) by using standard techniques of scatter ing theo-

ry, and is known as the Lippmann-Schwinger integral equation. Then the expression

of the re f lec t ion and transmission matrices can be obtained in terms of the matrix

(~_(k',k) from the def in i t ions (1.25) and (1.27), and the integrals (1.23) and

(1.24); they read

T(k) : 2ik ~2ik -~L (-k,k)] -1 . (1.30)

In the simple case with no discrete part of the spectrum the expression (!.29)

and the integral equation (1.28) clar i fy the relationship between the Fourier trans

form of the matrix Q(x)

Q(k) = dx exp(-ikx) Q(x) (1.31)

and the l inear approximation of the ST of Q(x)

R(N) = (2ik) -1Q(2k) ; (1.32)

however, by sett ing into the expression (1.29) the l inear approximation

~L(k' ,k) = ~ ( k ' + k ) , one obtains the improved approximation of the ST of Q(x)

RI I: I I I0>] (1.33)

Therefore the l inear integral equation (1.28) and the expression (1.29) are the ba-

sic equations not only to solve the d i rect problem, but also to invest igate appro-

ximate expressions of the ST; this second point, however, w i l l not be discussed any

fur ther.

In the same way as for the Fourier transformation,the app l i cab i l i t y of the ST

technique to solve evolu{ion equations re l ies on the i n v e r t i b i l i t y of the mapping

which associates the set of quant i t ies (1.20) to a matrix Q(x). That the Schr~din-

ger spectral problem defines a one-to-one correspondence between a large class of

"potentials" Q(x) and their corresponding ST: {R(k), - ~ < k < + ~ , p~J)," " p(J), ptJ)," "

j = 1,2 . . . . M} is one of the most important result of the scattering theory, which

44

has been discovered long time before i t s appl icat ion to nonlinear evolut ion equa-

t ions was recognized by GGKM. The solut ion of the inverse problem 1151, that is the

problem of reconstructing the matrix Q(x) from the knowledge of i t s spectral trans-

form, is then obtained by the fol lowing procedure: i ) compute the matrix

i+ : H (n) p(n) z) (1.34) M(z) = (27) -1 dk exp(ikz) R(k) + ~ p exp(-p (n) - n=l

then i i ) solve the Fredholm matrix integral equation

, / ÷ o o

K(x,x ' ) + M(x+x ' ) + I dy K(x,y) M(y+x ' ) = 0 , x _< x' i

J x (1.35)

and f i n a l l y i i i ) obtain Q(x) by the simple formula

Q(x) = - 2 d K ( x , x ) / d x , (1.36)

We notice that the integral equation (1.35), known as the Gel'fand-Levitan-Marchen-

ko equation~ is l inear , and that the variable x enters in th is equation only as a

parameter. Of course, as for the mapping Q(x) ÷ {R(k), p(J) , p(J) , P(J)} ,a lso the

inverse mapping {R(k), p(J), p(J), P(J)} ÷ Q(x), defined by Eqs. (1.34), (1.35) and

(1.36), is nonlinear; the matrix Q(x) w i l l be then understood to be the Inverse Spe~

t ra l Transform (IST) of {R(k), p~J)~' ' p(J) , p(J)} .

In analogy with what we are well used to do by means of the Fourier transforma-

t ion, we are now in the posit ion to reformulate problems concerning a matrix Q(x)

into the equivalent problems dealing with i t s ST and vilceversa; or, in other words,

to invest igate certain propert ies of Q(x) by looking at the corresponding proper-

t ies of i t s ST, and viceversa. In the fo l lowing, to be concise, we wi l l re fer to

x-space or to k-space in the cases we w i l l look at the Q(x) or at i t s ST, respect i -

vely; which one of these two representation is more convenient depends, of course,

on the problem being invest igated. An important instance of th is general rule w i l l

be given in the next Section by the nonlinear evolut ion equations, whi%h are much

more conveniently investigated in k-space than in x-space.

I t is remarkable that a special class of matrix valued functions Q(x) exists

such that the i r ST has a known e x p l i c i t analyt ic expression; these special Q(x)'s

are characterized by a ST with vanishing re f lec t ion coe f f i c ien t , to say of the type

{R(k) = O, p(J), p(J), P(J), j = 1 , 2 , . . . M}. Indeed, i f only the discrete part of

the ST is present, with M discrete eigenvalues, the kernel of the Gel ' fand-Levitan-

Marchenko equation is separable of rank M and this integral equation (1.35) reduces

45

to an algebraic equation. I t is an easy exercise to derive the simplest class cha-

racterized by just one discrete eigenvalue, namely by two posi t ive real parameters

p and p, and by a NxN one-dimensional projection matrix P; in fac t , set t ing

R(k) = O, M = I , p(1) = P, p(1) = p and p(1) = p in Eq.(1.34) and solving the co-

rresponding integral equation (1.35), one obtains the IST of {R(k) = O , p , p , P}

Q(x) = - {A/cos h 2 [ ( x - g)/x] }P (1.37)

where we have introduced the fol lowing new parameters

A : 2p 2 , x : p-1 , g = (2p) -11n(p/2p) (1.38)

in terms of the spectral parameters p and p, since they have a very transparent in-

terpretat ion in x-space, namely A is the amplitude, ~ is the width and ~ is the po-

s i t i on of the (single) minimum. I t is important to notice that the amplitude and

the width are related to each other in a def in i te way, i . e . A = 2/~ 2, and that the

matrix character of Q(x) is factorized in the x-independent projector P. For sake

of completeness, we now report the expressions that other quant i t ies, defined

above, take in this pa r t i cu la r l y simple, but important, case:

ex I 0,x+x, ] ex ,

T(k) = 1 + ~ i p / ( k - ip ) ] P- (1.42)

Let us now consider a one-parameter family of matrices Q(x,t) by introducing a

parametric dependence on a new variable t , from now on taken as the time. Then, of

course, a l l the corresponding quant i t ies we previously defined w i l l also depend on

time. Thus, as Q(x,t) evolves in time, the solutions of the SchrBdinger equations

(1.9) and (1.12) evolve in time and therefore, through the def in i t ions ( I .10a),

(1.12), (1.14), (1.17), (1.18) and (1.20), also the ST of Q(x,t) evolves in time:

{m(k, t ) , p ( J ) ( t ) , o ( J ) ( t ) , m(J)( t ) , j = 1,2 . . . . M(t ) } ; of course, mq.(1.1Ob) im-

pl ies as well a t-dependence of the transmission coef f fc ient T (k , t ) .

In analogy with the Fourier transformation and i t s appl icat ion to solve l inear

par t ia l d i f f e ren t ia l equations (see (1.1), (1.5) and (1 .4 ) ) i t is natural now to in

vestigate whether a time evolution exists such that, although possibly very compli-

cate in x-space, is simple in k-space. The f i r s t resul t of th is invest igat ion is

46

the s t r i k ing discovery of a class of nonlinear par t ia l d i f f e ren t i a l equations whose

solutions Q(x,t) are such that the i r corresponding Spectral Transforms sa t i s fy a

l inear ordinary d i f f e ren t i a l equation. By de f in i t i on , a nonlinear evolution equation

of th is class is solvable because i t s corresponding Cauchy problem can be solved by

means of l inear operations only. The way th is is accomplished is schematically

shown in the fol lowing " s o l v a b i l i t y diagram"

x-space k-space

solut ion at time t o DIRECT LINEAR PROBLErl spectral trans-

form of the solu- t ion at time t o

I Nonl i near Li near

evol ut i on evol ut i on

INVERSE LINEAR PROBLEM <

spectral trans- form of the solu- t ion at time t

solut ion at time t

In order to derive, via the generalized wronskian re lat ions technique, this

class of SNEE, we need few more def in i t ions which are tersely reported below. For a

given matrix Q(x), we introduce the fol lowing subsidiary Schr~dinger spectral pro-

blem, which, for the continuous part of the spectrum is defined by the d i f f e ren t i a l

equation

: # (x ,k) [Q(x ) -k21 , k ~ 0 (1.43) ~xx(X,k)

the NxN matrix solut ion #(x,k) being defined by the asymptotic conditions

#(x,k) x ÷ +--------~ exp(- i k x) + R(k) exp(i k x) (1.44a)

~(x,k) x ÷ _ j T(k) exp(-i kx) ~ (1.44b)

(1.43) d i f fe rs from (1.9) because now the matrix Q on the r .h .s , acts from the r igh t

rather than from the l e f t . The discrete part of the spectrum is characterized by

the vector d i f f e ren t i a l equation

j 12 . . . .

47

together with the normal izat ion condit ion

~(J) (x) ~(J (x) = 1 , j = 1,2 . . . . n , (1.46)

Here the so lu t ion ~(J)(x) is a N-dimensional row vector valued funct ion ( therefore,

in (1.46), the integrand is the usual scalar product), and i t s asymptotic behaviour,

because of the condit ion (1.46), uniquely defines the row vector c(J)

~(J) (x) x ÷ + =~ c(J)exp(- p (J )x ) , j : 1 ,2 , . . A (1.47)

A new spectral transform of Q(x), corresponding to th is new spectral problem, can

then be defined as

: { R(k), - ~ < k < + ~, p(J) , #(J) , P(J), j = 1 , 2 , . . . n } (1.48)

where, as before,

~(J) = E (j) E (j)+ > 0 , P(J) = (~( j ) ) - I E(j) + E(j), j = 1,2 . . . . M (1.49)

P(J) being the projection matrix corresponding to the nondegenerate eigenvalue

_~{j)2.- • However this other spectral transform (1.48) does not really add any new

piece to our mathematical machinary because we now prove that

ST of Q(x) : ST of Q(x) , (1.50)

We l i m i t our proof to the continuous part of the ST because, fo r the matrices Q(x)

sa t i s f y i ng the asymptotic condit ions (1.15), th is resu l t together with Eq.(1.16) im

p l ies the v a l i d i t y of (1.50); however the equa l i t y (1.50) holds fo r the la rger

class of matrices Q(x) considered in th is context. To prove that R(k) = R(k) we note

that the wronskian type expression

W(x,k) ~ ~(x,k) ~x(X,k) - ~x(X,k) ~(x,k) (1.51)

defines a x-independent NxN matr ix , namely

Wx(X,k) = 0 • (1.52)

Let us now evaluate the matr ix W by inser t ing in the d e f i n i t i o n (.1.51) the asympto-

t i c behaviour of the solut ions #(x,k) and ~ (x ,k ) , (1.44) and (1.10) respec t ive ly and

taking the l i m i t both as x ÷ - = and as x ÷ +

48

W(-~,k) = 0 , W(+~,k) = 2ik JR(k) - ~(k)] ~ (1.53)

this resul t , together with Eq.(1.52), implies that

R(k) = ~(k) (1.54)

which is jus t the equality (1.50) for the continuous part of the ST.

We f i na l l y notice that the hermit iani ty condition (1.6) implies

~(x,k) = [~(x , -k*) ] + (1.55)

and, through the asymptotic behaviour (1.10a) and (1.44a) together with the equa-

t ion (1.54), also

R+(k) = R(-K*) (1.56)

which, for real k, reduces to the equation (1.11) reported above. For completness,

we write also the (generalized) un i ta r i ty equation

?(-k) T(k) + R(-k R(K) = 1 (1.57)

which obtains by applying the same procedure given before to the following wrons-

kian expression

W_(x,k) z ~(x,-k) mx(X,k) - ~x(X,-k) ~(x,k) (1.58)

the detai ls of the derivation being l e f t to the reader as an exercise. For hermi-

t ian Q and real k Eqs.(1.55), (1.lOb), (1.44b) and (1.57) imply the well-known uni-

t a r i t y equation

T+(k) T(k) + R+(k) R(k) = I (1.59)

representing the par t ic le f lux conservation in a scattering process.

2. THE BASIC FORMULAE AND SNEE

The general method yielding al l the results contained in these lectures is ba-

sed on the ident i ty

49

i x

W(x2,k ) - W(Xl,k) = dx Wx(X,k) JX 1

(2.1)

where the NxN matrix W has the usual wronskian-type expression

W(x,k) = #l(X,k) ~2x(X,k) - ~ix(X,k) ~2(x,k) (2.2)

the equation (2.1), when special ized to the cases indicated below, produces the ba-

sic functional re lat ionship between Q's and the i r spectral transforms, which are

the bui lding blocks of the present approach.

Let ~ ' , respect ively ~, be two matrix solutions of Eqs.(1.43) (with potent ial

Q ' (x ) ) , respect ively (1.9), and consider the wronskian W defined by (2.2) with

~l(X,k) = ~ ' (x ,k ) , ~2(x,k) = F(X)~x(X,k) (2.3)

where F(x) is an arb i t ra ry (twice d i f fe ren t iab le ) matrix. Using on the r .h .s , of

the ident i t y (2.1) the d i f fe ren t ia l equations (1.43) and (1.9), and integrat ing

appropriately by parts Ii61, one easi ly gets the generalized wronskian iden t i t y

- + ~ ' F ~ + ~' F ~ - { ~'( -2k 2 F- Fxx+ FQ +Q'F)~ 2 ~ F~x x x x x

I x I [Ii ] : : + ~' dx' (FQ Q'F } ~' dx' (FQ - Q'F) ~x x Xo o

IX2dx ~' { - Fxx x + 2FxQ + 2Q 'F x - 4k2Fx + FQ x + Q~ F

Jx I

El x I J dx'(FQ Q'F) Q Q' ~ x - + dx' (FQ - Q ' F ) } ~ • x ° L x °

(2.4)

Note that x ° is an arb i t rary parameter. We are, of course, assuming F(x) and the po

tent ia ls Q(x) and Q'(x) to be su f f i c i en t l y regular to j us t i f y the various integra-

tions by parts required for the derivat ion of this equation. The key idea in der i -

ving the ident i ty (2.4) is to use the integrat ion by parts to end up with an inte-

grand matrix in the r .h .s , of a sandwich type, i .e . some matrix in between ~' and

~, while al l contr ibutions from the x-der ivat ive of i ' and ~ should come on!v ~:J~

the ends of the integrat ion in te rva l .

50

From (2.4) now we derive three important formulae:

i ) from the a rb i t ra r i ness of x o we get f i r s t of a l l the formula

E xI I - = dx #' { Q'M - MQ } ~ ( 2 . 5 ) ~'xM~ - i ' MY Xl Xl

where M is an a r b i t r a r y x-independent mat r ix . Using the asymptotic expressions

(1.10) and (1.44) where appropr iate, one f inds that the l i m i t x I ÷ - ~ of the

l . h . s , of (2.5) vanishes, whi le i t s l i m i t x 2 + + = is re la ted to the re f l ec t i on

coe f f i c ien ts corresponding to Q' and Q

2 i k n R ' ( k ) M - M R ( k ) ] = dx # ' ( x , k ) { Q ' ( x ) M - M Q ( x ) } T ( x , k ) , (2.6) -oo

This equation impl ies tha t , i f a matr ix M commutes with Q(x) for any x, [M,Q(x)] =0,

M commutes with R(k) fo r any k, [M,R(k)] = O.

i i ) Next we set F(x) = N, where N is x-independent, and, choosing x o =+ ~ and

taking the l i m i t s x I + - = and x 2 + + ~,we get

Q,(x) LQ'(x')N- NQ(x' [Q'(x')N- x dx' ) ] - I x dx' NQ(x') ]Q(x) } ~(x,k) (.2.7) f

i i i ) F i na l l y , we consider a matr ix F(x) , that is a r b i t r a r y , but vanishes

asympto t ica l l y . S p e c i f i c a l l y , we assume

F(+=) = Fx(+-=) = Fxx(+- =) = 0 (2.s)

and, again with x o = + ~ and taking in (2.4) the l i m i t s x I ÷ - ~ and x 2 + + ~, we

obtain

(2 ik) 2 ]_ d x # ' ( x , k ) Fx(X) ~(x ,k) : ~ ' ( x , k ) { Fxxx(X) - 2Q'(x) Fx(X) -

2Fx(X) Q ( x ) - Q # ( x ) F ( x ) - F ( X ) Q x ( X ) - Q ' ( x ) dx' ' ( x ' ) F(x ' ) - F (x ' )Q(x ' ) +

~ Y X

dx' Q' (x ' F(x ' ) - F(x ' ) Q(x ') Q(x) }T ( x , k ) (2.9)

X

51

The content of th is formula is more transparently shown by introducing the integro-

d i f f e ren t i a l l inear operator A, defined by the fol lowing formulae that detai l i t s

action on a generic matrix F(x) (vanishing at + ~):

I ~ d x ' AF(x) = Fxx(X) - 2L'(x)FQ F(x) + F(x) Q(x) 4] + I ~ F(x') • (2.10)

r+oo

x Ix X rF(x) : Qx ( ) F(x) + F(x) Qx(X) + ' (x ) Q (x ' ) F(x ') -

- Q'(x) F(x ' ) Q(x' ) - Q ' (x ' ) F (x ' ) Q(x) + F(x ' ) Q(x' ) Q(x)I (2.11) ]

Thus, i f F(x) is an arb i t ra ry matrix (vanishing as Ixl ÷ ~), the equation (2.9) can

be rewr i t ten in the more compact form

i+ _ dx ~ ' (x ,k ) {AF(x) }~(x ,k ) = (2ik) 2 _ dx ~ ' (x ,k ) F(x) ~(x,k) (2.12)

(since here F(x) is a rb i t ra ry , no confusion should be caused by replacing the ma-

t r i × F x entering in (2.9) with F(x), which is now the generic matrix in (2.12)). I f

we now assume that the matrix F(x) vanishes asymptotically with al l i ts derivat ives

l im r~|dnF(x)/dxn| = 0 , n : 0,1,2 . . . . . (2.13) L 4 X # + o o

we can le t A act repeatedly on F(x), obtaining

I+: 0x I:: ~ ' (x ,k ) {AnF(x) }~ (x ,k ) = (2ik) 2n dx ~ ' (x ,k )F(x )~(x ,k ) , n=0,1 ,2 . . . . ;

(2.14)

i t should be emphasized that in this formula, as well as in al l the s imi lar ones

given below, the operators never act on the wave funct ions, even though one of

these is always wr i t ten on the r .h .s . (and could not be wr i t ten on the l . h . s . ,

since one is generally dealing with nonco~nuting matrices). F inal ly , from (2.14)

there immediately follows

i +~ r+~

d x ~ ' ( x , k ) { f ( A ) F ( x ) } ~ ( x , k ) = f ( -4k 2) j ~ ' (x ,k ) F(x) ~(x,k) (2.15) _ _ d x

where f (z ) is an arb i t ra ry ent i re funct ion. This remarkable equation, in which

F(x) is essent ia l ly a rb i t ra ry , except for the requirement (2.13), should be

somehow seen as analogous to

52

(2.16)

which is well-known in Fourier analysis.

The basic formulae of th is approach obtain by inser t ing F(x) = Q'(x)M - MQ(x),

respect ively F(x) = rN, in (2.15) and by using on the r .h .s . Eq.(2.6), respect ively

Eq.(2.7). In th is manner we get

2i k f ( -4k 2) FR ' (k )M-MR(k) ] = L .]

(2ik)2g(-4k2)[R'(k)N+NR(k)] =

Iildx ,217, l+co

j dx ~ ' (x ,k ) {g(A) FN }~ (x ,k ) , (2.18) - c o

These equations provide the main tool of our treatment. Let us reuemphasize that

they have been shown to hold for a rb i t ra ry (ent i re) funct ions f and g, and for ar-

b i t r a ry x-independent matrices M and N. The operators A and F are defined by Eqs.

(2.10) and (2.11); they involve the two matrices Q(x) and Q' (x) , to which there co-

rrespond the re f l ec t i on coef f i c ien ts R(k) and R'(k) .

Thus the equation (2.17) and (2.18) re late the two matrices Q'(x) and Q(x) to

the continuous part of t he i r ST (see d e f i n i t i o n (1.20)) . Simi lar equations connec-

t ing Q'(x) and Q(x) to the discrete part of t he i r ST ( i . e . to the discrete spectrum

of t he i r corresponding Schr~dinger eigenvalue problems), as well as to t he i r co-

rresponding transmission coef f i c ien ts T ' (k) and T(k) , can also be obtained by the

generalized wronskian re la t ion technique; we do not report them here since, for our

present purposes, they are not rea l l y needed (and the interested reader can f ind

them in reference 1111).

We now proceed to discuss the f i r s t of several important appl icat ions of the

basic formulae (2.17) and (2.18), namely the der ivat ion of the class of SNEE. To

th is aim, we introduce a basis of the N2-dimensional space of NxN matrices, namely

the N 2 hermitian matrices a , ~ = 0,1 . . . . . N 2 - 1 , with a o = 1; greek indices run

over the values 0 ,1 ,2 , . . . ,N 2 - I ; l a t i n indices over the values 1,2, . . . . N 2 - 1 . The

convention that repeated indices are summed is always understood. Furthermore, the

conventional notat ion for commutators and anticommutators used throughout is

[A,B] = AB - BA, {A,B} = AB + BA.

We introduce a one-parameter family of matrices Q(x , t ) , the parameter t being

the time, and we set

53

Q(x) : Q(x,t) , Q'(x) : Q(x,t+At) (2.19)

that of course also imples

R(k) = R(k,t) , R'(k) = R(k,t+At) (2.20)

and analogous relat ions for the parameters of the discrete spectrum and for the

transmission coef f ic ients .

Now we inser t this ansatz in Eqs. (2.17) and (2.18), and invest igate the l im i t

At ÷ O, obtaining thereby a set of relat ions that provide the basis for es tab l is -

shing a class of nonlinear evolution equations solvable by the ST associated to the

matrix Schr~dinger problem.

Let us begin by wr i t ing down the operators A and r, defined by Eqs.(2.10) and

(2.11), respect ively, in the l im i t Q'(x) = Q(x). We call these operators L, respec-

t i ve l y G; they are defined by deta i l ing the i r action on a generic matrix F(x) (va-

nishing at i n f i n i t y ) : ÷oo f

LF(x) = Fxx(X) - 2{Q(x, t ) , F(x)} + G I dx' F(x ') (2.21) )

X

I E Ix<> GF(x) = {Qx(X,t), F(x)} + F [Q(x, t ) , dx' F(x ') • (2.22)

x

Then set M = o n in Eq.(2.17), and wr i te fn in place of f ; for At

y ie lds

2ikfn(-4k2) [on,R(k, t ) I = I i l d x ~ ( x , k , t ) { f n ( L ) [~n,Q(x,t)~ } ~ ( x , k , t ) .

= O, this

(2.23)

Had we set instead M = ~o' we would have obtained for At = 0 the t r i v i a l ident i t y

0 = O; therefore, in order to obtain a nontr iv ia l equation, we set M = {o/At in

Eq.(2.17), and take the l im i t At ÷ O. This y ields

2 i k f ( -4k 2) Rt (k , t ) = I dx # (x ,k , t ) { f(L) Qt(x, t ) } ~ ( x , k , t ) < (2.24)

Set f i n a l l y , in Eq.(2.18), N = o and wr i te g~

y ie lds

in place of g. For At = O, this

54

(2ik) 2 gu(-4k 2) {o ,R(k , t ) } = # (x ,k , t ) :{g (L)G ~ } ~ ( x , k , t ) , (2.25)

Note that the indices n and u in Eq.(2.23) and in Eq.(2.25), respect ively, may or

may not be understood as summed upon, since the functions fn and gn' due to the i r

arb i t rar iness, might a l l but one chosen to vanish. Furthermore, the arb i t rar iness

of f , fn and gu in the 3 equations wr i t ten above implies that these functions could

also parametrical ly depend on t ; they must of course be independent of x.

From Eqs.(2.23), (2.24) and (2.25), the following result is immediately im-

plied: i f the matrix Q(x,t) satisfies the nonlinear equation

F 7 f ( L , t ) Qt : an(L't) ~n '~ + B (L,t) G o (2.26)

the corresponding re f lec t ion coe f f i c ien t R(k,t) sa t is f ies the l i nea r ordinary d i f f e

rent ia l equation

f(-4k2't) Rt : an(-4k2't) [~n'~ + 2ik ~ ( -4k2, t ) {~ , R} . (2.27)

In the following, we restrict, for simplicity, our attention to the case f : l ,

and moreover we assume the (entire but otherwise arbitrary) functions a and # to n

be independent of t. Thus we focus attention on the SNEE

Qt ( x ' t ) an(L) n' Q(x,t) + ~ (L)G ~

The nonl inear i ty of th is equation or iginates from the dependence of the operator L

on Q, as exp l i c i t e l y shown by i ts de f in i t i on Eqs.(2.21) and (2.22). In order to

solve the i n i t i a l value problem for the nonlinear evolut ion equation (2.28), accor-

ding to the so l vab i l i t y diagram we have previously discussed, we need to be able to

solve the corresponding i n i t i a l value problem for the ST of Q(x, t ) . For the cont i -

nuous part of the ST we have the fol lowing ordinary l inear matrix d i f f e ren t ia l

equation

Rt(k , t ) = an(-4k 2) ~n ,R(k , t ) ] + 2ik ~ (-4k 2) {~ ,R(k, t ) } (2.29)

which obtains by specia l iz ing Eq.(2.27) to the present subclass of SNEE.

As for the time evolut ion of the discrete part of the ST of Q(x, t ) , one might

apply the generalized wronskian method as i t has been done for the re f lec t ion coef-

f i c i e n t ; however, a more straightforward procedure, y ie ld ing the same evolutilon

equations, even though i t applies only in the case of matrices Q that vanish asympt~

55

t i c a l l y faster than exponential ly, takes advantage of the relat ionship between the discrete spectrum parameters p£J)" " and c ( j ) and the locations and residues of the poles of the ref lect ion coef f ic ient (see Eq.(1-16)).

For the evolution of p (J ) ( t ) , being k = ip (J ) ( t ) a simple pole of R(k, t ) , one immediately obtains from (2.29)

p~J)(t) : 0 , j : 1,1 . . . . M (2.30)

To wri te down the evolution equation of the remaining part of the ST, i t is convenient to introduce the matrices

Ic c(J)(t) = c (j) (J)(t) = p(J)(t) P(J)(t) , j = 1,2 . . . . M (2.31)

since they are simply related to the residue of R(k,t) at k = ip(J)(t); then for these matrices, Eq.(2.29) implies

c~J)(t) = ~n(4P ( j)2) [~n,C(J)(t)] - 2p ( j ) ~ (4p ( j)2) {~ , c (J ) ( t ) } (2.32)

The constancy of the eigenvalues _p(j)2, i .e . thei r time independence, as im- plied by Eq.(2.30), shows that the flow (2.28) is isospectral, a character ist ic feature of many important evolution equations discovered so far 1171.

We are now in the posit ion to prove the so lvab i l i t y of the nonlinear equation (2.28). In fact , given the i n i t i a l value 1181

Qo(X) = Q(x,O) (2.33)

we solve the direct problem, which is l inear, and compute the ST of Qo(X)

ST of Q(x,O) : {Ro(k), p~J), p~J), P~J), j : 1,2 . . . . M} (2.34)

then, we integrate the l inear d i f fe rent ia l equations (2.29), (2.30) and (2.32), with the i n i t i a l condition (2.34), and obtain the exp l i c i t solutions

R(k,t) = exp [4ik Bo(-4k2)t] exp{t [an(-4k2)+ 2ik Bn(-4k2)] ~n } '

{ t [-~n(-4k 2) + 2ik Bn(-4k2)] ~n } ' (2.35) Ro(k) exp

P(J)(t) = P~J) (2.36) u

56

c(J)(t) = p(J)(t)P(J)(t)= p~J)exp [-4p (j) Bo(4p(J)2)t].

e x p { t [~n(4P ( j)2) - 2p ( j ) Bn(4p(j)2)] ~n } P~J)"

e x p { t I-~n(4p(J)2) - 2p ( j ) ~n(4p(j)2)] ~n } ' (2.37)

and therefore the ST at the time t. Final ly, by insertion of (2.35), (2.36) and

(2.37) into Eq. (1.34), we solve the l inear Gel'fand-Levit a n-Marchenko inteqral

equation (1.35) of the inverse problem and obtain through Eq.(1.36), the solution

Q(x,t) of the nonlinear evolution equation (2.28) at the time t .

The great simpl i f icat ion obtained by transforming the nonlinear evolution

equation (2.28) into the corresponding l inear evolution equations for the ST (2.29),

(2.30) and (2.32) is the f i r s t good example of the following "golden rule": to in-

vestigate a SNEE of the class (2.28) (or, for this matter, of the class (2.26))and

the properties of i ts solutions, go from x-space to k-space where the problem is

l inear. Of course, the main d i f f i cu l t y in solving the Cauchy problem is found in

performing the transformation from the x-space to the k-space and from the k-space

back to the x-space, these transformations being defined through the solution of

the direct and inverse problems associated to the matrix Schroedinger operator. We

emphasize that, in addition to being l inear, the evolution equations in the

k-space have also the virtue of not coupling to each other the d i f ferent components

of the ST; thus, the time evolution of the parameters p and C (or p and P, see

(2.31)) characterizing each discrete eigenvalue is quite independent of the possiNe

presence of the continuous component, or of other discrete eigenvalues. As i t wi l l

be clear in the next section, i t is this fact, coupled with the solvabi l i ty through

the ST method, that originates the remarkable behaviour of "sol i tons", in part~

cular thei r s tab i l i t y . I t also motivates the interest in the single-sol i ton solu-

t ion, since th is, not only provides a remarkable special solution of the class of

SNEEs being investigated, but indeed constitutes a component (that may, or may not,

show up asymptotically; see below) of a broad class of solutions.

We end this section wri t ing down in our formalism few examples of nonlinear

evolution equations which are already well-known 151:

i ) the Korteweg-de Vries equation

ut(x,t) = Uxxx(X,t) - 6u(x,t) Ux(X,t) (2.38)

obtains by setting N=I , to say the matrix Q(x,t) = u(x, t) reduces to a scalar

5?

f i e l d , and ~o(Z) = z/2.

i i ) the modif ied Korteweg-de Vries equation

q t ( x , t ) = qxxx(X, t ) + 6q2(x , t ) qx (X , t ) (2.39)

obtains wi th N =2, by w r i t i n g the 2 x 2 matr ix Q(x , t ) in terms of the scalar func- t ion q ( x , t ) according to the fo l lowing ansatz

q2(x,t) qx(X,t q ( x , t ) = (2.40)

~ q x ( X , t ) - q 2 ( x , t ) J '

and by choosing an(Z) = Bn(Z ) = 0 (n =1,2 ,3) and Bo(Z) = z/2 .

i i i ) the nonl inear Schr6dinger equation

i ? t ( x , t ) = - ~ x x ( X , t ) - 2 1 ~ ( x , t ) l 2 ~ ( x , t ) (2.41)

obtains wi th N =2, B (z) = O, ~1(z) = ~2(z) = O, ~3(z) = iz /2 and re la t i ng the 2 x 2 matr ix Q(x , t ) to the funct ion ~ (x , t ) as fo l lows

Q(x , t ) = ~ -l~(x't)12

i v ) the sine-Gordon equation

%(x,t) 2~ -l~(x,t)l

@xt(X,t) : s in [ # ( x , t ) ]

obtains wi th N=2, an(Z ) = ~n(Z) = 0 ( n = 1 , 2 , 3 ) , ~o(Z) = (2z) - I and

2x X,t// v) the boomeron equation 1191

• (2.42)

(2.43)

(2.44)

U t ( x , t ) = b . Vx (x , t ) (2.45a)

58

Vxt(X,t) = Uxx (X , t ) b+aA~x (X , t ) - 2Vx (x , t )A LV(x,t) A (2.45b)

is a system of coupled nonlinear equations for the scalar f ie ld U(x,t) and the

three-dimensional vector f ie ld V(x,t) , and obtains with N =2, ~o(Z) = O,

an(Z) = an/2i and Bn(Z) = bn/2 (n = 1,2,3) being constants and defining the two

three-dimensional vectors ~ and b, and

dx' Q(x',t) = U(x,t) + o .V(x , t ) (2.46)

where ~ ~ (~1' ~2' ~3 ) are the three Pauli matrices.

Nothe that a rigorous treatment of the important equations i i ) , i i i ) and

iv) requires an extension of the present formalism to non hermitian 1201 matrices

Q(x,t). Although th~s can be easily done, in the following we wi l l consider only

cases where the hermitianity condition (1.6) holds. Therefore, since the basis ma-

trices ~ are hermitian, suff icient conditions to guarantee that Q(x,t) remain her-

mitian throughout i ts time evolution i f i t is hermitian at the i n i t i a l time t o , are

that the functions B and ~ in Eq. (2.28) be real and imaginary respectively n

* = B* (2.47) a n =-an , Bv

In the next section the motion of the so l i tons, pa r t i cu la r l y the boomeron equa

t ion, w i l l be invest igated, and the i r non vanishing acceleration w i l l be exhibi ted.

3. SOLITONS

To set up the proper language to describe the nonlinear phenomena modeled by

the nonlinear evolution equations solvable by the ST method, the continuous part of

the ST wi l l be hereafter referred to as the background component of the ST. The

reason is that a solution of a SNEE, whose ST has no discrete part, evolves in time

very simi lar ly to a wave-packet undergoing the well-known dispersion phenomenon. On

the other hand, we wi l l refer to the discrete part of the ST as to the soliton com-

ponent; in fact, solutions with a vanishing reflection coefficient, i .e. with

ST = {R(k) = O, p(J) (J) (J) , p , P , j = 1,2 . . . . . M } ,

behave in a quite d i f fe ren t way, since a sort of balance between dispersive and non-

l inear ef fects give them s t a b i l i t y and l o c a l i z a b i l i t y properties whilch show up in i n

terest ing and beauti ful physical phenomena. An enthusiast ic descript ion of a pheno-

menon of th is sort in hydrodynamics, has been given by J. Scott-Russel 1211 in the

59

far 1844, th is being an ear ly in t roduct ion of sol i tons to observational science,

well before the rather recent theoret ica l descr ipt ion. An important rSle in disco-

vering so l i tons and understanding the i r properties has been played by computer expe

riments, among which the most crucial have been those produced in 1965 by N.J. Za-

busky and M. Kruskal 122,231 , who also introduced the name "so l i t on " . Since then

the number of papers devoted to so l i tons has been growing up to now in an explosive

way, and an up-do-date review of so l i ton phenomenology in some branches of physics

has been presented by R.K. Bullough 131.

Here we focus our discussion only on those solut ions of the nonl inear evolu-

t ion equation (2.28) whose ST contains only the so l i ton component~ namely with a

vanishing background component.

Let us s ta r t with the more fami l ia r 1-sol i ton so lu t ion in the simpler case

N = I , so that eq.(2.28) reduces to the scalar f i e l d equation

Qt (x , t ) = 2~o(L) Qx(X,t) (3.1)

the easiest way to construct th is solut ion is to consider i t s spectral transform at

a given time t , to say, M= I , R(k,t) = O, p (1) ( t ) = p ( t ) , p (1) ( t ) = p, and apply

the techniques described in the previous sections. The corresponding inverse spec-

t ra l transform was already obtained solving the Marchenko equation, and given by

the expression (1.37) which in the present case reads

Q(x,t) = -A/cos Ix- (3.2)

where we have taken in to account the expressions (1.38) of the amplitude A and

width ~ of the so l i ton in terms of the spectral parameter p, and the fact that th is

parameter p is constant because of the isospectral evolut ion expressed by the equa

t ion (2.30). Therefore the t-dependence of the solut ion Q(x, t ) enters only through

the funct ion

~(t) = (1/2p) In ~(t)/2p] (3.3)

and is immediately obtained by specializing the general solution (.2.37) to the scalar case, namely

g(t) = go-2Bo (4p2) t (3.4)

With these f indings in our hands, we should draw our a t tent ion on the fo l lowing

facts: i ) the amplitude A and the width x of the so l i ton do not depend e i ther on

time or on the par t i cu la r SNEE belonging to the class (3.1) , and are related to

60

each other

A = 2/I 2 ; (3.5)

i i ) the function ~( t ) should be interpreted as the posit ion of the sol i ton at the

time t and the expression of the spectral parameter p( t ) in terms of the sol i ton mo

tion is

p(t) - 2p exp [2p~(t~ ; (3.6)

i i i ) the soliton does not accelerate and behaves as a free particle moving with the

constant speed

v = -2 Bo(4P 2) (3.7)

This part ic le- l ike behaviour has certainly motivated the interest of many physi-

cists which consider this mathematical framework, namely nonlinear theory, as a

f i r s t step towards an elementary particle theory which avoids the unpleasant point-

l ike model; however, this is only but one of many others, and important, applica-

tions of the concept of soliton to physics.

In order to investigate which kind of soliton dynamics is implied by these non-

l inear evolution equations, we have to investigate the large I t l evolution of a man~

-soliton solution. After the numerical experiments showing a two-soliton col l ision

carried out by N.J. Zabusky 1221 , the f i r s t proof that two coll iding solitons even-

tual ly reappear unchanged is due to P.D. Lax 161 (they both investigated the KdV

equation). Let us consider then a solution containing only M solitons, i ts spectral

transform being {R(k,t) = O, p~J)(t)," " p~J]," " j =1,2 . . . . M}; although at f i n i t e time

the nonlinear effects may prevent us to recognize the individual sol i tons, they

wi l l ul t imately show up well separated from each other because of thei r d i f ferent

veloci t ies v ( j ) = -2 Bo(4P ( j )2) (see Eq.(3.7)). Indeed, the well-known s tab i l i t y

of the sol i ton is nicely expressed by the asymptotic behaviour of the M-soliton so-

lut ion for large values of I t l , which shows M 1-soli ton bumps with the usual shape

Ix ( J ) ( t ) ] / 1 ( j ) } ~ (3.8) Q~J)(x,t) = -A(J)/cosh 2 { -~+ , as t ÷ ±

The effect of the i r nonlinear interact ion is accounted for by the following simple

rule ~ J ) ( t ) = ~ !J ) ( t ) + d ( j ) , d ~ J ) ( t ) / d t = v ( j ) =-2Bo(4p(j)2) (3.9)

the amplitudes and widths having the usual expression A ( j ) = 2p ( j)2 and 1(J)=l/p ( j ) Therefore, while the amplitudes, widths and speeds have not been changed by the in- teract ion, the only dynamical effect is the displacement d ( j ) of the j - t h sol i ton

61

posit ion with respect to i t s i n i t i a l free motion. The expression of the displacement

d ( j ) in terms of the sol i ton widths ~(J) 's , obtained independently 1241 ( for the

KdV equation) by V.E. Zakharov (1971), M. Toda and M. Wadati (1972) and S. Tanaka

(1972), shows that these displacements are given by the sum.of M-I two-soliton con-

t r ibu t ions , and this proves that sol i tons in teract only via two-body forces. Fur-

thermore the relat ionship sat is f ied by the displacements d ~j),' " namely M

(d(J)/~ ( j ) ) = 0 is natura l ly understood as the free motion of the center of j = l mass of the M-soliton system. The qua l i ta t i ve picture emerging from these results

is tha't of an isolated system of M classical par t ic les. The a t t rac t i ve character of

the two-soliton force is easi ly shown in the case of a two-soliton co l l i s ion ; i f

v (1) > v (2), the sol i ton 1 is advanced while the other one is delayed, the i r corres

ponding sh i f ts being

d(1) = ~(1) i n l ( x (Z )+x (2 ) ) / ( ~ (Z )_x (2 ) ) i >O , d (2 )=x(2 ) in l (x ( l )_x (2 ) ) / (x (1 )+~(2) ) 1

< 0 (3.10)

With the aim of f inding sol i tons with a r icher dynamics ( for instance, so l i -

tons with more degrees of freedom interact ing via a t t rac t i ve or repulsive forces),

l e t us look at a system of coupled nonlinear evolution equations for a multi-compo-

nent f i e l d , namely for a matrix Q(x, t ) . We have already shown in the previous sec-

t ion that important nonlinear equations such as the sine-Gordon, nonlinear Schr~din

ger and modified KdV equations can be derived, as special cases of non hermitian

2x2 matrices, from the class of SNEE (2.28). However also in these cases the inte~

acting sol i tons experience in the co l l i s ion only a displacement of the i r posit ion;

in terest ing features are nevertheless exhibited by the sine-Gordon equation which,

among other vir tues ( for instance, i t s r e l a t i v i s t i c invariance), has solutions des-

cr ib ing two kinks (sol i tons) co l l id ing via repulsive interact ion as well as a kink-

-ant ik ink bound state 1251 ("breather" or "bion"). In order to explore the general

properties of the sol i ton solutions of the equation (2.28), we consider f i r s t the

case of N xN hermitian matrix f ie lds .

Let us focus f i r s t on the 1-sol i ton solut ion. This is easi ly obtained inser-

t ing in the Marchenko equation (1.35) i t s spectral-transform {R(k,t) = O, p, p ( t ) ,

P( t ) } . This resu l t reads

Q(x,t) = - {A/cos h2{ Ix - ~ ( t ) ] l ~ } } P(t) . (3.11)

Again the constant (scalar) A and the width ~ are simply related to the spectral pa

rameters, A = 2p 2 and ~ = p- l . The posit ion of the sol i ton at time t :~ ( t ) is re la -

ted to spectral parameter p(t) by the formula (3.6).

62

The novelty of th is solut ion is due to the remarkable fact that ~( t ) is gene-

r a l l y not l inear in t , i . e . this sol i ton generally moves with a speed which varies

in time. The mechanism which is responsible for th is in terest ing feature originates

from the time dependence of the projection matrix P(~), and i t s re la t ion to the so-

l i t on speed (see below). This implies that one of the N 2 components of our f i e l d so

lu t ion , say a matrix element Q i j ( x , t ) , although i t has the usual (cosh) -2 shape in

the x var iab le, generally has the t-dependent amplitude A P i j ( t ) which makes them

look d i f fe ren t from the usual sol i tons. Our sol i ton therefore is characterized by

i t s posit ion ~( t ) and by the one-dimensional projection matrix P(t) that shall be

occasionally referred to as the polar izat ion of the so l i ton.

The e f fec t of the polar izat ion time-dependence on the motion of the sol i ton is

better investigated considering the time evolution of the corresponding spectral

transform; i f c ( t ) is the N-dimensional vector such that c ( t )c+( t ) = p( t ) P(t) (see

Eq.(2.31)), then Eq.(2.32) and Eq.(3.6) imply the fol lowing equations of motion for

the sol i ton posit ion and polar izat ion

~ t ( t ) : - 2 ( c ( t ) , B c ( t ) ) / l l c ( t ) I I 2 - 2~o(4p2) (3.12)

Pt( t ) = [ (~ -2p~) , P(t) ] - 4p P(t) 611 - P(t) ] (3.13)

where we have set

z ~n(4P 2) ~n ' # z Bn (4p2) ~n " (3.14)

The SNEE's discussed by Wadati and Kamijo 1121 imply ~ = 0 and require that ~ com-

mutes with P( t ) ; therefore in this case Eqs.(3.13) and (3.12) describe the usual s~

l i t on with P(t) = P(O) and ~ t ( t ) = ~t(O). A sol i ton with constant speed can be a so

lu t ion also of our more general equations (3.12) and (3.13) i f i t s i n i t i a l polar iz E

t ion P z P(O) is a solut ion of the matrix equation o

[(~ - 2pB), Po] - 4p Po ~(i - Po ) 0 (3.15) P

I f th is is the case, P(t) = Po is the solut ion and the ve loc i ty of the sol i ton t u r n

out to be constant. Indeed, i f the uni t vector u(t) is introduced

u(t ) z c( t ) l l lc ( t ) l l , P(t) : u(t) u+(t) (3.16)

then i t is eas i ly found that the solutions of the matrix equation (3.15), wich are

projectors on one dimensional subspaces, are given by the fol lowing eigenvalue equ E

t ion

63

Po = UoU+o ' (~ - 2p B) u o = uu 0 (3.17)

+ This implies that , i f [~,B] = O, being B = #+ and ~ = -~ (see (2.47)and (3.14)) ,

the matr ix ~ - 2p~ is normal and i t s N eigenvectors u k, k = 1,2 . . . . N, define N l - so

l i t on solut ions with constant speed v k = -2(u k, BUk) - 2Bo(4P 2) and polar izat ion Pk = UkU~ " Of course these solut ions are very special and, as i t w i l l be clear in the fo l lowing, very unstable with respect to small changes of the i n i t i a l value of

the polar izat ion matr ix. In order to discuss the general behaviour of our so l i tons ,

we give the e x p l i c i t so lut ion of the equations of the motion (3.12) and (3.13)

~(t) = C O -2Bo(4p2)t + (1/2p) In[(Uo,E(t)Uo) ] (3.18)

with

E ( t ) - {exp [ t ( ~ - 2pB)]} + {exp [ t ( ~ - 2pB)] } • (3.19)

The corresponding formula for P(t) is

P(t) = [(Uo,E(t)Uo)] -1 exp I t ( s - 2 p B ) ] Po exp I t ( s - 2 p B ) +] (3.20)

+ where , o f c o u r s e , u o = u(O) and Po = UoUo = P(O). To d i s c u s s t he b e h a v i o u r o f t he

so l i ton i t is convenient to introduce the spectral decomposition of the hermit ian matr ix E ( t ) :

N E(t) : Z exp [2p ~k(t)] Ek(t) (3.21)

k=l

where, of course,

Ek(t) E l ( t ) = 6klEk(t) (3.22)

Note that the quantities ~k(t), as well as the projection operators Ek(t), depend

on the functions a n and B n that characterize the structure of the SNEE under consi- deration; they depend on the in i t ia l conditions only through the value of p. The time evolution of ~(t) and P(t) depends moreover on the in i t ia l vector u o z u(O), through the quantities

ek(t) ~ (u o, Ek(t) u o) (3.23)

The explici t formulae are

N ~(t) = C o - 2~o(4p2)t + (I/2p)In{k!lek(t)exp[2 p ~k(t)] } , (3.24)

64

N P(t) ={ ~ e k ( t ) e x p [ 2 P ~ k ( t ) ] } - l . e x p l t ( ~ - 2p8) ] Po.eXp [ t(m - 2p8)+I

k=l (3.25)

In order to understand the effects on the sol i ton motion due to the matrices ~ and

8, l e t us consider f i r s t the special case 8 = Oo Then E(t) = I , ~(t)=~o-2PSo(4p2)t;

the sol i ton moves with constant ve loc i ty , while i t s polar izat ion matrix P(t) evolv~

according to the unitary transformation exp(tm), namely i t s corresponding uni t vec-

tor u( t ) undergoes a precessional motion ( th is because ~ is ant ihermit ian) . Since

the t-dependence due to 8 o is r e l a t i v e l y t r i v i a l , in the fol lowing we set for sim-

p l i c i t y 8 ° ~ O. I f , on the contrary, the matr ix 8 is not vanishing, the sol i ton ex-

periences on acceleration and i t s motion may be very complicate; however i t is ra-

ther easy to discuss i t s asymptotic motion, i . e . for I t l ÷ =, i f we assume that

[m,~ : 0 (3.26)

In this case the matrix E(t) does not depend on the matrix ~ and therefore, as i t

is implied by Eq.(3.18), the matrix ~ has no ef fect on the posit ion of the sol i ton

at any time; the matrix ~ contributes only to the time evolution of the projector

P(t) by superimposing to the motion due to the matrix 8 the "precession ef fect "

given by the uni tary transformation exp(t~). Furthermore th is e f fect due to the ma-

t r i x ~ on the polar izat ion P(t) disappears as I t l ÷ ~ (see below) so that the asym~

to t i c motion of the sol i ton depends only on the matrix 8. Indeed, in this case,

Ek(t) = E k , ~k(t) = -2t8 (k) , k = 1,2 . . . . N (3.27)

where the 8(k) 's , which are the eigenvalues of the matrix 8, are ordered according

to the prescr ipt ion 8 (1) < 8 (2) < . . . < 8 (N) (assuming no degeneracy), and the pro-

ject ion operators E k are those projecting on the eigenvectors of the matrix 8. The

quant i t ies ek(t ) are t-independent, and, i f we exclude the special solutions mentio

ned above which correspond to the i n i t i a l condition ek(O) = 6kl, the sol i ton moves

at large I t l as a free par t i c le

~( t ) =~o + ( I /2p ) In [em(O) ] - 2ts(m)+ O{exp [ -4 t (8 (re+l) ~ 8(m))] } (3.28a)

a s t ÷ t ~

a s t ÷ - =

where the indeces m and M are such that

(3.28b)

m < M , ek(O ) : 0 i f k < m , k > M (3.29)

65

and in general m = I and M=N. I f B ¢lj~ ~ >O>~mJ~ ~, the so l i t on moves towards the r i g h t

as t ÷ + ~, and i t recedes also to the r i gh t as t ÷ - ~. This behaviour j u s t i f i e s

the in t roduc t ion of the term "boomeron". I t is a lso easy to prove that in th is case,

the po la r i za t i on matr ix has the asymptotic value

l im P(t) = E m , l im P(t) = E M (3.30) t ÷ + ~ t ÷ -

where again the indeces m and M are def ined by Eq.(3.29).

In the general case, the behaviour of the so l i t on may be completely d i f f e r e n t .

However, i t can be proved tha t , i f the so l i t on escapes to i n f i n i t y , i t can do so at

most wi th a speed tha t is asympto t ica l ly constant. I f i t is not so, and that th is

may be the case, i t is demonstrated by the example reported below, the so l i t on ins-

tead o s c i l l a t e s i n d e f i n i t e l y , and th is one we w i l l re fe r to as a " t rappon".

The s t ra i gh tes t way to d isp lay e x p l i c i t e l y the "boomeron" and "trappon" beha-

v iour is to consider the simplest novel SNEE of the c lass, namely the boomeron equa

t ion (2.45). C lear ly the quan t i t i es U(x, t ) and V (x , t ) s a t i s f y the asymptotic condi-

t ions

U(+~,t) = Ux(±~,t) = 0 , V(+~,L) = Vx(±~,t) = 0 o (3.31)

In view of the remarkable features of the s o l i t o n so lu t ions described below, the

discovery of a physical phenomenon modeled by the boomeron equation should be consi

dered a very i n te res t i ng goal.

In the present vector no ta t ion , the po la r i za t i on of the so l i t on w i l l be repre-

sented by the un i t vector ~( t ) def ined by the fo l l ow ing equation

P(t) = (1/2)(1 + ~(t).~) (3.32)

Then the 1-soliton solution with amplitude A and width ~ (see Eq.(3.11)) reads

U(x,t)=-(2A)l/2.{1+exp{2[x-~(t)]/~} }-I , V(x,t)=U(x,t) fi(t) . (3.33)

The position and polarization evolve in time according to the equations 1261

- - + A AS] ~t(t) (3.34a)

~t(t) = -6.A(t) (3.34b)

The explicit solution of these equations is

~(t) = ~o+ (~/2) In [n+E+(t) +n_E_(t) + sS(t) + cC(t)l (3.35) L ]

66

n( t ) : [ n + E + ( t ) + n _ E _ ( t ) + ~ S ( t ) + ~ C ( t ) ] / [ n + E + ( t ) + n _ E _ ( t ) + ~ S ( t ) + c C ( t ) ] (3.36)

E l ( t ) : exp(mav_t ) , S( t ) : s i n ( a v + t ) , C(t) = cos(a~+t ) (3.37)

, [ ] ~± = z { z 2 cos 2 O + (z 2 1)2/4 1/2 ± (z 2 1 ) /2 } I / 2 , z = ~a/(2b) ,

cos 0 = ~.b (3.38)

±

~_(aAn o) $ nz- lv+(6Ano ) - z - l ( aA~ ) ) } / (V2+~2_ ) , n = sign (A.b) (3.39)

2)_ , ._>. - " +

= n o n+ - n_ (3.40)

-~ : [nv_(~.~o) - z - I u+(~.~o) ] / ( 2 + 2),_ c : 1 - n + - n_ . (3.41)

In the simpler case AA6 = 0 (we always use the notat ion { = v v) we f ind that i f

the i n i t i a l po la r i za t ion ~(0) = no coincides with one of the two un i t vectors

n+ = sb, then we have the two special so lu t ions : n( t ) = -b, g ( t ) = go + bt and

n( t ) = b , ~ ( t ) = ~o - bt. I f n(O) ~ ±b , ~( t ) ~ b and ~( t ) ~ -b fo r a l l f i n i t e

t , whi le for the asymptotic motion we f ind ~(±~) = ~± = $ b , gt(±~) = ± b; th is

shows a boomeron coming from i n f i n i t y with the ve l oc i t y -b and going back to i n f i -

n i t y with opposite ve l oc i t y . This motion is wel l character ized by the fo l lowing

theorem: the boomeron coordinate g( t ) coincides with that of a ( n o n r e l a t i v i s t i c )

pa r t i c l e of un i t mass, with i n i t i a l pos i t ion g(O) = go and i n i t i a l speed

~t (0) = -~ 'no ' moving in the external potent ia l ~ ( ~ - ~o ) , with

@(X) = (i/2) b 2 11-(b.~o )2]exp(-4x/~) . (3.42)

In the general case, excluding the special one (A.6) = 0 that is discussed separa-

t e l y below, i f the i n i t i a l po la r i za t ion no coincides with one of the two un i t vec-

tors

n+ = ($n~+a ~ z -1 ~ 6 - z - l ~ A b ) / ( l + 2)_ (3.43)

we have the two special so lut ions

n ( t ) = ~+ _ , ~ ( t ) : ~o -+ v t (3.44)

describing a sol±ton moving with a conStant speed of modulus

67

v = bzu_ = b{ ~z 2cos28 + ( 1 / 4 ) ( 1 - z 2 ) 2 ] I /2 + ( 1 / 2 ) ( 1 - z 2 ) } 1/2 " (3.45)

Otherwise both ~( t ) and ~ t ( t ) vary with t ime; at na f i n i t e time 6( t ) coincides with

n+ or ~_, whi le asymptot ica l ly

~(±~) = ~+_ , ~t(±~) : i v (3.46)

Note that th is impl ies tha t , independently of the condit ions assigned at time t =0

(provided 6 o ~ ~÷),_ both in the extreme future and in the remote past the boomeron

escapes, or recedes, toward the r i g h t , wi th the same asymptotic speed

~ ( t ) - (~o + v I t l ) = 0 [exp(-2a~_ I t l ) ] , as t ÷ ± ~ (3.47)

Note also that the asymptotic speed v (see Eq.(3.45)) is a monotonic funct ion of

the dimensionless parameter z (defined by Eq.(3.38)) , th is parameter being propor-

t iona l to the width of the sol±ton; the speed v can take i t s value only in the f i -

n i te in te rva l b cos 8 ~ v ~ b, approaching i t s maximum value b fo r a very narrow so

l i t o n (~ ÷ O) and i t s minimum value b cos 8 fo r a very f l a t one (~ ÷ ~).

In the special case ~.b = O, a c r i t i c a l value of the width ~ ex i s t s , which is

x = 2b/a (corresponding to z = l ) , such that a sol±ton with X < x behaves qui te c c d i f f e r e n t l y from a sol±ton with x > Xc" Indeed, i f the i n i t i a l po la r i za t ion coin-

cides with e i the r one of the un i t vectors (O(x) = 1 i f x>O, 8(0) = 1/2, O(x) = 0

i f x < O)

~± : - z - l { ± ( z 2 - I ) 1/2 g ( z - 1 ) ~ i z ( 1 - z 2 ) 1/2 8 ( l - z ) b + [z 2 g ( l - z ) + 8 ( z - 1 ) ] AAb} (3.48)

i t does not change with t ime, 6 ( t ) = 6 o, and the sol±ton does not move at a l l i f

z ~ 1, whi le i f z < 1 i t moves with constant speed

~( t ) : C o ± v t i f 6 o = n+ _ , v = b(1 - z2) 1/2 (3.49)

I f instead the i n i t i a l po la r i za t ion 6 ° coincides ne i ther with 6+ nor with 6_, the

po la r i za t ion does change with time: i t never coincides with ~+ nor ~_, but, i f z5

i t tends asymptot ica l ly to the values ~+ ( i . e . fi(±~) = A+), whi le i f z > l i t pre-

cedes pe r i od i ca l l y , with period

T = (2= z /a) (z 2 - 1) -1/2 • (3.50)

As fo r the behaviour of the coordinate C( t ) , i t is best understood noting that i t

68

coincides with that of a par t ic le of uni t mass, with i n i t i a l posit ion ~(0) = t o

and i n i t i a l speed i t(O) = -~'no' moving in the external potential 9 ( ~ - t o) with

~(x) = (1/2)b 2exp (-2x/k) { [ ( [ 2 + ¥ 2 + z x ] 2 + z 2 2)/ ( 2 + x2 ) ] exp(-2x/k) -

- 2 z ( z + y ) } (3.51)

where ~ = (A.~o) and ¥ = (AAb.no). The tota l energy of this par t ic le has the sim-

ple expression

E : 9(0) + (1/2)(~.~o)2: (1/2)v 2 s ign( I -z ) (3.52)

with v defined by Eq.(3.49). Thus i t is posi t ive i f z < l , in which case we have a

boomeron (see f i g . ( 1 ) ) which escapes, or recedes, to the r igh t as t ÷ ± ~, moving

asymptot ical ly with the constant speed v. When the tota l energy is negative, for

z > l , th is so l i ton, that we name trappon (see f i g . ( 2 ) ) , behaves as a par t ic le

trapped in the potent ia l , osc i l l a t i ng i nde f i n i t e l y around the equi l ibr ium posit ion

~o ÷ Xm' where x m is the value of x where the potential (3.51) takes i t s single ne-

gative minimum. This osc i l l a to ry motion is periodic with the period given by the

formula (3.50). The marginal case z =1 corresponds to the motion of a zero-energy

par t ic le that always escapes (or recedes) to posi t ive i n f i n i t y , but with a va-

nishing asymptotic ve loc i ty . We note, however, that trappons can ex is t only in th is

case (a.b) = O.

I t should be emphasized that , for a given SNEE of type (2.45) with (~.6) = O,

in a many-soliton solut ion a l l the types of sol i ton behaviour described above may

be simultaneously present, depending on the values of the width ~ ( in par t icu lar ,

whether or not i t exceedes the value 2b/a) and of the i n i t i a l polar izat ion no of

each so l i ton.

The addit ional degrees of freedom of our sol i tons due to the i r polar izat ion

cer ta in ly make the i r dynamics r icher than i t was for the usual sol i tons. Indeed, an

M-soliton solut ion w i l l break up, as t + ± ~, in M ~ M bumps with the usual one-

so l i ton shape, M- M being therefore the number of trappons contained in th is solu-

t ion. The interact ion among these sol i tons, due to the nonl inear i ty of the evolu-

t ion equation, should manifest i t s e l f in the asymptotic value of the polar izat ions

of the (asymptot ical ly) free sol i tons which show up at large I t ] ; in addit ion to

th i s , one should also f ind a displacement of the j - th so l i ton asymptotic posit ion

with respect to the undisturbed one-soliton motion, namely, with respect to the

function t£J)( t ) ' " entering into the spectral transform through the spectral parame-

ter p£J)(t)" " according to the formula (3.6). As an example of these processes, we

b r i e f l y report the main results on the boomeron-boomeron and boomeron-trappon co l l i

sions. Let (R(k, t) = O, p~J), p~J)( t ) , P~J)(t) , j = 1,2} be the ST of the two-sol i -

89

ton solutions.

Assume f i r s t that ~(1)( t ) , P(1)(t) and ~(2)( t ) , P(2)(t) are two boomeron-type solutions of the evolution equations (3.12) and (3.13), and let us introduce their asymptotic behaviour

[ ] ! _(k)+vkt k : 1 ,2 (3.53a) lim ~ ( k ) ( t ) - ~ ! k ) ( t ) = 0 , ~ k ) ( t )=~o - ' t ÷ ± ~ - -

lim P(k)(t) = ~(k)v_+ , k =1,2 • (3.53b) t÷±oo

To be def in i te, let boomerOn 1 be asymptotically faster than boomeron 2, i .e. v 1>v2>0, As t + m ~, the two-boomeron solution 1(see f ig . (3 ) ) shows two wellr sepa rated soli tons, which freely move with position ×~k)(t) and polarization II~ k), each of them being characterized by the usual expression

_(k) (3.54) Q~k)(x,t) : - { A ( k ) / c o s h2{ [ x - x ~ k ) ( t ) ] / x ( k ) } } H ±

where A (k) = 2p (k)2 and x(k) = 1/p(k). The effect of the col l is ion can be descri- bed as follows: the asymptotic motion and polarization of the (asymptotically) fas- ter boomeron are not changed at al l

instead, for the (asymptotically) slower boomeron we find an asymptotic displace- ment

x(2) ( t ) =C!2) ( t )+d+, d+=- (1 /2)~(2) ln [1 -4~(1)~(2) t r (p ! l ) p!2))]

and the following symptotic polarization

(3.56)

= exp(2d±/x(2)){ (~( I ) /~ (2) ) [ l_exp(_2d±/~(2) ] p~ l )+ p~2)

2p(1){p~1), p~2)}} (3.57)

(3.58a)

d + = d d = - (1/2) ~(2) In{1-2~ (1) u (2) [1 + z (I) z (2) +

( i - z ( i ) 2 ) 1/2 ( I - z ( 2 ) 2 ) 1/2] }

where we have set u(k) ~ p(k)/(p(1) + p(2)). Specializing this result to the boome- ron equation with (~.b) = 0, we obtain

70

~2 ) (1/2)(1 ~ 2 ) . ~ ) , ^(2) exp(2d/X(2)) { (u(1) /u(2) ) = + m± =

[ ] ^ ( 1 ) + (1-2~ (1)) ^ (2 ) } (3.58b) 1-2~ (2)-exp(-2d/~ (2)) n± n±

with an obvious meaning of the symbols (see Eqs.(3.32), (3.38), (3.48) and (3.49)).

Assume now that the two-sol±ton solut ions of the boomeron equation (with (a.b) = 0) describes a boomeron-trappon system and that C(1)( t ) , P(1)(t) and ~ (2 ) ( t ) , P(2)(t) are the two solutions of the evolution equations (3.34a,b) corres- ponding to the boomeron and to the trappon, respectively; namely z ( I ) <1 and z(2)>1. In this case (see f i g . ( 4 ) ) , for large I t l , one finds one sol±ton moving towards or from i n f i n i t y , and one trappon moving in a confined region. Therefore the asympto- t i c behaviours (3.53a) and (3.53b) apply now only to the boomeron ( i .e . for k =1), while the function ~2)( t ) ' ' and the projection matrix P~2)(t)' ' are periodic functions

of t , with period T = 2~ z(2)/[a(z (2)2- 1)I/2]. As t + ± =, we find that the boo-

meron bump is given by the expression (3.54) (for k = I ) , with A (I) = 2p ~I)2,' ' x( 1 ) = 1/p £1)' ' , and its asymptotic motion and polarization are given by the formula

(3.55), to say that the coll ision with the trappon does not affect the asymptotic

motion of the boomeron, which therefore behaves for large I t l , as i f i t never met

the trappon on its way. On the contrary, the asymptotic motion of the trappon is

not that of an undisturbed trappon, namely that characterized by the position

~2)(t)" ' and polarization P(2)(t) as for the one-trappon solution discussed above.

Now the asymptotic trappon, as t ÷ ± ~, is

Q~2)(x,t) = - { A ( 2 ) / c o s h 2 { [ x - × ~ 2 ) ( t ) ] / ~ (2)} } ~ 2 ) ( t ) (3.59)

with A (2) = 2p(2)2, x(2) = 1/p(2) and

X!2)( t ) = ~(2) ( t ) + ~+(t) ,

a±(t) = -(1/2)~ (2) I n { 1 - 4 u (1) ~ t r P ( t ) } (3.60a)

R~2)(t) = exp [2a±( t ) /~ (2 ) ] { (~ (1 ) / u (2 ) ) ( l _ex p [ - 2 a ± ( t ) / x ( 2 ) 1 ) p ~ l ) +

p(2)( t ) _ 2 ~ ( 1 ) { p ~ 1 ) p (2) ( t ) } } (3.60b)

We notice that the functions ~+(t) and the polar izat ions ~+-(2)(t) are again periodic in time, with the same period T = 2x z(2) / [a(z (2) 1) 1/2 ]- .Few words are appro- pr iate to comment on these f indings. F i rs t , we note that the center of mass of the two-sol±ton system does not f reely move as for the KdV equation, because here the

71

sol i tons, in addition to the i r in teract ion, move in the i r corresponding "external

potent ia l" (3.51). The second point concerns the strange asymmetry between the two

co l l id ing boomerons, namely boomeron 1 does modify the asymptotic motion of boome-

ron 2 but not viceversa; the same asymmetry occurs in the boomeron-trappon c o l l i -

sion, This feature of the co l l i s ion processes described by the boomeron equation

originates from the asymmetric boundary conditions (3.31) which imply the simple

rule: the asymptotic motion and polar izat ion of each sol i ton is affected only by

those other sol i tons which asymptotical ly move on i t s r igh t side, but not by those

sol i tons which asymptotical ly remain on i t s l e f t side.

4. BASIC NONLINEAR EQUATIONS ANDRELATED RESULTS

In sect.2 the basic formulae (2.17) and (2.18) have been investigated in the

l i m i t Q'(x) ÷ Q(x), thus deriving the class of SNEE's. This is jus t one of several

results implied by these basic formulae. Other resul ts , concerning important pro-

pert ies of the solutions of these SNEE's, obtain without taking the l i m i t

Q'(x) ÷ Q(x) and w i l l be tersely presented here, To s impl i fy th is presentation and

to focus only on the main ideas, throught this section we w i l l avoid the matrix for

malism considering only the scalar f i e l d case (N =1); the general ization I I i i to

the matrix case, although essential to invest igate important nonlinear evolution

equations, does not contribute to our general understanding of this matter.

We focus on the formulae for the re f lec t ion coef f ic ients , since they are suff i l

c ient to obtain a l l the interest ing f ina l resul ts. The derivat ion of analogous rela

t ionship for the transmission coeff ic ients can be done by a closely analogous tech-

nique; we omit also to report on the analogous results for the discrete-spectrum pa

rameters that can be s im i l a r l y derived by the generalized wronskian method, or from

the results given below for the re f lect ion coef f ic ient by assuming the v a l i d i t y of

eq.(1.16) (one should be aware, however, that some f ine points deserve a more de-

ta i led treatment).

Let us f i r s t rewrite down the two basic equations (2.17) and (2.18) in this

simpler (scalar) case f+oo

(2ik)2g(-4k 2) - e e

I>x, (Q ' (x ) -Q(x ) ) ( Q ' ( x ' ) - Q ( x ' ) ) } (4.2)

72

where Q(x) and Q'(x) are real functions vanishing at i n f i n i t y (see sect. I ) , and

R(k) and R'(k) are the i r corresponding re f lec t ion coef f ic ients ; f (z ) and g(z) are

a rb i t ra ry (ent i re) function and A is the fol lowing l inear in tegro -d i f fe ren t ia l ope

rator depending on Q(x) and Q'(x) . l+oo

x (+~

0,x, t x,i -

JX X ~

The basic nonlinear equations obtain d i rec t l y form eqs.(4.1) and (4.2), since they

imply that, i f the two functions Q(x) and Q'(x) are related by the formula

[ [ f(A) Q'(x) - Q(x) + g(A) Qx(X) + Qx(X) + (Q (x) - Q(x))

j dx' (Q ' (x ' ) - Q(x') ) ] = 0 (4.4)

x

the corresponding re f lec t ion coef f ic ients R(k) and R'(k) are related by the equa-

t ion

The importance of formula (4.4) is that i t displays in closed form the e x p l i c i t , i f

complicated, re lat ionship between two potent ials Q(x) and Q'(x) whose corresponding

re f lec t ion coef f ic ients are related by the simple l inear formula (4.5).

I t should be emphasized that the functions f and g might also depend on other

variables (such as t , see below); they must of course be independent of xo

We now use these important f indings as a tool to invest igate the properties of

the solut ion of a SNEE of the class (3.1) with the i n i t i a l condition

Q(x,O) = Qo(X) (4.6)

eq.(3.1) obtains, of course, from (2.28) for N = 1, or, i t can be d i rec t l y derived

from our basic equation (4.4) by choosing Q'(x) = Q(x,t+At), Q(x) = Q(x, t ) ,

f ( z ) = 1 /At , g (z ) = - go(Z) and t a k i n g the l i m i t B t ÷ O. The l i n e a r o p e r a t o r L i s

defined here as

LF(x) : Fxx(X) - 4Q(x,t) F(x) + 2Qx(x,t) dx 'F(x ' ) ~ (4.7) Jx

73

Let us regard then solution of the Cauchy problem (3.1) and (4.6) as a "poten-

t ia l " which depends parametrically on the time variable t, on the in i t ia l condition

Qo(X) and on the function 6o(Z) characterizing a particular SNEE

Q(x) : q(x,t,Qo,6 o) (4.8)

A systematic application of the basic equations (4.4) and (4.5) is based on conside

ring Q' as obtained from Q by changing only t (the result being a resolvent formu-

la), or by changing only Qo(X) (the result being a B~cklund transformation) or by

changing only Bo(Z). More general transformations are obtained 1271 combining to-

gether these three transformations.

i ) B~cklund transformations 1281: Assume that in eq.(4.4)

q ' (x) = Q(x,t,Q~,B o) , Q(x) = Q(x,t,Qo,~o) (4.9)

then (applying our golden rule) le t us write the relevant equations in the k-space, namely

R(k,t) = exp [4ik6 o ( -4k2) t ] Ro(k) (4.10)

R'(k,t) : If(-4k2)-2ikg(-4k2)]/[f(-4k2)+ 2ikg(-4k2)] R(k,t) (411)

therefore from these equations i t is clear that the assumption (4.9) is verified i f

the functions f and g are independent of t. Indeed, in this case, R'(k,t) is rela-

ted to R~(k) ~ R'(k,O) by the same equation (4.10) that relates R(k,t) to

Ro(k) z R(k,O). But such a time evolution for the reflection coefficient R'(k,t)

corresponds to the time evolution (3.1) for the corresponding potential Q'(x,t).

Conclusion: i f Q(x,t) satisfies the SNEE (3.1), and Q'(x,t) is related to Q(x,t)

by (4.4), with f and g independent of t , then Q'(x,t) satisfies the same SNEE

(3.1).

This class of B~cklund transformations is interest ing because i t holds for the

whole class(3.1)ofSNEE's, there being no rest r ic t ion on the functions f and g that

characterize the B~cklund transformation. The condition that a real solution Q(x,t)

should be transformed in a real solution Q' (x , t ) by (4.4) is just the requirement that f(z) and g(z) be real analyt ic:

f (z) = f * (z* ) , g(z) = g*(z*) • (4.12)

74

A res t r i c t i on on f and g i s , however, implied by the requirement that both Q

and Q' have the properties that were assumed to begin with; in par t icu lar the pro-

perty that , i f the function Q(x) is f i n i t e valued for real x and vanishes asympto-

t i c a l l y (as x ÷ ± ~), the function Q' (x) , related to Q(x) by (4.4), also has these

propert ies; but, while i t would be d i f f i c u l t to specify such a res t r i c t i on on the

basis of eq.(4.4), i t is actual ly quite easy to in terpret i t on the basis of eq.

(4.5), since i t then amounts to the requirement that the simple mu l t i p l i ca t i ve fac-

tor

@(k) = [f(-4k2)-2ikg(-4k2)]/[f(-4k2)+ 2ikg(-4k2)] (4.13)

does not bring about any unacceptable property of the reflection coefficient R'(k).

This means, for instance, that @(k) (that is generally a meromorphic function of k)

should not have poles of higher order than the f i rs t , or even simple poles coinci-

ding with poles of R(k), or poles in disallowed regions fo the complex k-plane

(such as the region Re k ~ O, Im k > O, i f the potentials are required to be real

and to vanish asymptotically faster than exponentially).

Let us proceed here to an analysis of the simpler B~cklund transformation,

that corresponds to both f and g being constants. I t is convenient to characterize

by the constant p = ½ f/g, so that (4.4) reduces to the one-pa this transformation

rameter subclass of transformations

R' (k , t ) : - [ ( k + i p ) / ( k - i p ) ] R(k,t) (4.14)

and the B~cklund transformation reads

Q ' (x , t ) = Q(x,t) - (2p) -1 I Q ~ ( x , t ) + Q x ( x , t ) + ( Q ' ( x , t ) - Q(x,t) )

x d x ' ( Q ' ( x ' , t ) - Q ( x ' , t ) ) (4.15)

To discuss these B~cklund t r ans fo rmat ion and re la ted r e s u l t s , i t is convenient

to work in terms of the in tegra l of Q, ra ther than Q i t s e l f . We therefore introduce

the function

j: f W(x,t) = dx 'Q(x ' , t ) , W'(x,t) = d x ' Q ' ( x ' , t ) (4.16) X

that c lear ly sa t i s fy the boundary conditions

i W(+~,t) = Wx(±~,t) : W'(+~,t) = Wx(±~,t) = 0 (4.17)

75

and from which the potent ia ls can of course be recovered through the formula

Q(x, t ) = - Wx(X,t) , Q ' ( x , t ) = - W~(x,t) • (4.18)

Indeed i t is often convenient to wr i te also the SNEE's (2.28) in terms of W rather

than Q; for instance, i t is in terms of W (compare (2.46) wi th (4.16)) that the boo

meron equation 1201 (2.45) can be wr i t ten as a pure par t ia l d i f f e r e n t i a l , rather

than i n t e g r o - d i f f e r e n t i a l , nonlinear equation.

In terms of W and W' the B~cklund transformation (4.15) becomes

1 Clearly th is equation is invar ian t under the transformation W ~ W',

p ~ -p; and th is is consistent with the corresponding transformation (4.14) in

k-space, that is c lear ly invar ian t under R ~ R', p ~ -p. Thus the B~cklund

transformation with parameter -p can be considered the inverse transformation to

that with parameter p. Note, however, that general ly (4.19) wi th a given W (consis-

tent with (4.17)) can be integrated to y ie ld a W' consistent with (4.17), only i f

p>O; and indeed in such a case the boundary condit ions (4.17) are sa t is f ied auto-

mat ica l ly , so that there s t i l l remains a certain arb i t rar iness in W', since the

"constant of in tegrat ion" is not f ixed by the boundary condit ions (4.17). Note that

th is "constant of in tegrat ion" does not appear in the transformation (4.14) of the

re f lec t ion coe f f i c i en t and therefore i t mu~be related to the discrete (so l i ton)

component of the ST of Q ' ( x , t ) ; consistent ly th is "constant of in tegrat ion" implies

a dependence on t , that is d i f f e ren t for the d i f f e ren t equations of the class (3.1)

and that can be ascertained only by inser t ing Q' in the SNEE i t s e l f , thereby obta i -

ning an equation, invo lv ing only the time var iable, for the " in tegrat ion constant".

This s i tua t ion is c lear ly connected to the fact that the formula (4.19) is often

referred in the l i t e ra tu re as "one ha l f " of a B~cklund transformation; for the deri

vat ion w i th in our formalism of the other "one ha l f " , for the KdV equation, see K.M.

Case and S.C. Chiu 129 I. Therefore, the picture implied by (4.14), and our previous

analysis, is tha t , by solv ing (4.19) for W', one gets general ly a so lu t ion Q ' = - W' X

having one more so l i ton (corresponding to the discrete eigenvalue k 2= -p2) , than

the solut ion Q = -W x (so l i ton creat ion) ; th is also implies that , for a given Q ha-

ving a discrete eigenvalue for k = i q , q>O, eqs.(4.19) and (4.17) would exceptio-

na l l y be solvable for W' even for a negative value of p, namely for p =-q; the co-

rresponding solut ion Q' would then have one less so l i ton that Q (so l i ton ann ih i la -

t i on ) .

I t is i ns t ruc t i ve to solve eq.(4.19) in the special case W=O. One then obtains

for W(x,t) , the s ingle so l i ton solut ion (3.2) , namely

76

W'(x, t) : - 2p {1 - tgh [ p ( x - ~ ) ] } (4.20)

the constant of integration ~depends of course on t ; i ts exp l i c i t time evolution

may be ascertained by insert ing (4.20) in the SNEE (3.1) and by solving the resul-

t ing equation, that invo lv~only the variable t , recovering thereby for ~( t ) the e~

pression (3.4).

An important implication of

mations (4.4) (and therefore the

the expression (4.5) is that al l B~cklund transfo~

subclass of them (4.15), or, equivalently, (4.19))

commute; i t should be emphasized that this property, that is essent ial ly t r i v i a l

when interpreted through (4.5), is instead highly nontr iv ia l when viewed in the co L

text of (4.4) (or even (4.19)), due to the nonlinear structure of these formulae.

An important consequence of this property is the so-called "nonlinear superposition

pr inciple". This can be derived in the following way: le t Qo(x,t) be a solution of

SNEE (3.1), Ql(X,t) respectively Q2(x,t) , the solution~ ~ of the same SNEE relatedz ~ to

i t by the B~cklund transformation (4.15) with p = p~l j , respectively p = p~2j,

Q1o(x,t) the solution related to Q1(x,t) by the B~cklund transformation (4.15) with

p = p' J, and Q21(x,t) the solution related to Q2(x,t) by the B~cklund transforma-

t ion (4.15) with p = p(1). Then, with obvious notation, we get from (4.14)

R12(k,t)={{(k+ip(2))(k+ip(1))]/[(k-ip(2))(k-ip(1))~ }Ro(k,t) (4.21a)

R21(k, t ) : { [ ( k + i p ( 1 ) ) ( k + i p ( 2 ) ) I / [ ( k - i p ( 1 ) ) ( k - i p ( 2 ) ) ] }Ro(k, t ) (4.21b)

that c lear ly imply

and therefore also

R12(k,t ) : R21(k,t ) (4.22)

Q12(x,t) = Q21(x,t) (4.23)

Let us write out the formulae corresponding to the statements that we have

jus t made, working again with the more convenient quantit ies W

Wlx + Wox = - (1/2)(W 1-Wo)(4p(1) + W 1 - W o) (4.24a)

W2x + Wox = - (1/2)(W 2-Wo)(4p(2) + W 2 - Wo) (4.24b)

W12 x+Wlx = - (I/2)(WI2-W1)(4p(2) +W12-W1) (4.24c)

W12 x + W2x = - (I/2)(W12-W2)(4p(1)+W12-W2) (4.24d)

77

where we have taken into account that W12 = W21, as implied by (4.23) and (4.16).

With a l i t t l e algebra, we now eliminate al l d i f ferent ia l terms, getting thereby the formula

W12(x,t) : Wo(x,t )_2 (p (1 )+p(2 ) ) [Wl(X,t)_W2(x,t) I

[2(p(i) _ p(2)) + Wl(X,t ) _ w2(x,t)]-I (4.25)

that provides, through (4.16), an exp l i c i t expression of the solution Q12 of the

SNEE (3.1), in terms of an arbi t rary solution Qo and of the two solutions Q1 and Q2

related to Qo by the simple Backlund transformation (4.15).

Among the implications of a formula such as (4.25), we merely mention here

that generally QI2 has two more solitons than Qo' and that, s tar t ing from Qo =0'

as shown f i r s t , for the KdV equation, by Wahlquist and Estabrook 1311, one can use

(4.25) to generate the whole ladder of mult isol i ton solutions. For instance, i t can

be ver i f ied that the two-soliton so l u t i ons

r k ~ p

W(x,t) : - 2(p (1 )+p (2 ) ) (1 -T IT2 ) - I (T I+=2 -2T IT2 )

( k ) ( p ( 1 ) + p ( 2 ) ) - l { 1 - tgh {p(k) [x - ~ (k ) ( t ) I } } , k : 1,2,

(4.26a)

(4.26b)

which can be easily obtained solving the Marchenko equation (1.35) with R(k,t) = 0 and p(k)( t ) = 2p(k)exp[2p (k) ~ (k ) ( t ) ] , k=1,2, is recovered by insert ing in (4.25),

together with W o = 0, the two solutions

Wl(X,t) = -2p(I) {I -tgh{p (I) [x-~(1)(t)] + 6}} (4.27)

W2(x,t) = - 2 p ( 2 ) { i - cotgh {p(2) Ix - ~ (2 ) ( t ) l + 6 } } (4.28)

where 6 = (1/2)In [(p(2) + p(1))/(p(2)_ p(1))] (4.29)

and p(2) > p(1) > 0 (4.30)

Note that i t is this last inequali ty that distinguishes the d i f ferent roles played

by the solutions W 1 and W2; indeed i t can be easily shown that only the choice (4.27) and (4.28) yields a nonsingular solution W12.

Since al l the results we have given can be extended in a straightforward way

to the matrix f ie lds 1111, we wi l l now br ie f ly discuss, as a second remarkable appl i

78

cation of the B~cklund transformations, the conservation laws which are sat is f ied

by the solut ions of the boomeron equation 1191 (2.45). The technique described here

to obtain the conserved quant i t ies is closely analogous to the resul ts f i r s t given

for the KdV equation 1311.

We f i r s t note that the scalar equation (2.45a) has already the form of a con-

servation law. Then we show how to obtain from i t an i n f i n i t e sequence of other

conservation laws by exp lo i t ing the dependence on the parameter p of the new f i e ld

variables U' and 7 ' , obtained from U and V by a Backlund transformation, of the

type discussed above, characterized by p.

As implied by the formula (4.14), which holds also in the matrix case, the

uni t transformation is obtained in the l im i t p ÷ ~; therefore, in order to discuss

a B~cklund transformation in the neighbour of the iden t i t y transformation~ i t is

more convenient to introduce instead the parameter

E = - (2p) -1 • (4.31)

The expression for th is one-parameter family of B~cklund transformations is found

to be

[ ] = [ ' + U x + ( 1 / 2 ) ( V ' - V ) 2 ] (4.32a) (U' -U) 1 - (E/2)(U' -U) a U x

(7' - ~) 1 - a(U' - U) = a(V' x + Vx) (4.32b)

But, of course, also U' and V' sa t is fy the conservation law (2.45a)

U~(x,t) = b.V~(x, t ) (4.33)

which, on the other hand, contains a parametric dependence on a, as implied by

(4.32). Introducing now in the Backlund transformation (4.32) the asymptotic expan-

sion N

U' = U + ~ a n U (n) + 0(~ N+z) (4.34a) n=l

~, : ~ + N an ~(n) + o(EN+I) (4.34b)

n=l

one obtains the recursion re lat ions

n-1 (n-m) ~(m) ~(n-m)) n > u(n+1) . (n) + (1/2) ~ (u (m)u + , - 1 (4.35a) : u x m= 1

79

n i l u(m) ~(n-m) > ~(n+l) = ~(n) + , n _ 1 m=l

(4.35b)

with the i n i t i a l conditions

7 (1) = 2V (4.36) U(1) = 2Ux ' x

I t is easily seen that these relat ions, together with (3.31), imply

)( ÷(n)( U (n ±~,t) : 0 , V ±~,t) : 0 , n ~ 1 (4.37)

Inserting now (4.34) in (4.33), we obtain the in f in i te sequence of conservation Iaws

u~n)(x,t) = ~.v~n)(x, t ) , n ~ 1 (4.38)

that, together with (4.37), imply that the quantit ies

Cn = (1/4) I i l d x u (n ) (x , t ) , n ~ 1 (4.39)

are constants of the motion. I t turns, however, out that the quantit ies u(2n)(x, t )

are perfect derivatives of quantit ies that vanish asymptotically, so that C2n = O.

The odd-numbered C2n+1 do instead y ie ld nontr ivial conserved quanti t ies, for ins- tance:

C 1 : -(1/2) U(-~,t) (4.40a)

÷2 C 3 = (1/2) _ dx U2x(X,t) + Vx(x,t ) (4.40b

r+~ +2 j Vxx,X, ,]

I t can be easily ver i f ied that these constants of the motion have a simple expres-

sion in terms of the 2x2 matrix f ie ld Q(x,t) related to the scalar and vector

f ie lds U(x,t) and V(x, t ) by (2.46); for instance, the expressions (4.40) then read

(4.41b j -~

f _

t r { [ ~ d x { 2 Q(x,t) 3 (4.41c)

8O

For the s ingle-sol i ton solution (3.33),or (3.11), these conserved quanti t ies take the simple expressions C I = ~- I , C3 = (4/3) X-3, C5 = (16/5) ~-5.

i i ) Resolvent formula: Assume 1181 now that in eq.(4.4)

Q'(x) = Q(x,t,Qo,B o) , Q(x) = Q(x,O,Qo,B o) (4.42)

then in the k-space eq.(4.5) should be read with

R'(k) = R(k,t) , R(K) = R(k,O) = Ro(k) (4.43)

But then eq. (4.5) becomes consistent with the actual time evolution (4.10) i f we choose

f (z2 , t ) = cosh [z ~o(Z2)t] , g(z2,t) = - z - l s i n h Iz Bo(Z2)t] (4.44)

Therefore the corresponding transformation in x-space

cosh [A 1/2 Bo(A)t I [Q(x , t ) -Qo(X l -A-1 /2s inh [A1/2BO(A)t] { Qx(x,t) +

relates a function Qo(X) just to the function Q(x,t) into which Qo(X) has evolved at time t according to the SNEE (3.1) (the i n i t i a l condition (4.6) being t r i v i a l l y ver i f ied by sett ing t =0 in (4.45), and A being the operator (4.3) with Q'(x) and Q(x) replaced by Q(x,t) and Qo(X) respect ively). For this reason we name the func- t ional equation (4.45) the resolvent formula.

An interest ing, i f complicate, operator ident i ty obtains by choosing

2Bo(Z) = v , vt : a , Qo(X) = f (x) (4.46)

therefore, solving eq.(3.1),

Q(x,t) = f ( x + a ) (4.47)

and (4.45) becomes the following operator iden t i t y

cosh (A I/2 a/2) [f(x+a) - f (x ) ] - A - I /2 sinh (A 1/2 a/2) { fx (X+a)

If(x+a)- f(x)] I~°°dx ' If(x'+a)-f(x')] } =0

+ fx(X) +

(4.48)

81

where f (x) is an arb i t rary function, except for the requirement that i t vanishes

asymptotically, and A is the operator + ~

JX

! Lf(x+a) - f (x ) ] 1 dx' dx" f (x ' +a) - f ( x ' ) F(x") (4.49) J X X l

Note that the l inear izat ion of this equation yields the elementary operator ident i - ty

f ( x+a ) = exp(a d/dx) f (x) • (4.50)

i i i ) Transformation between solutions of d i f ferent SNEE's Assume f i na l l y that in eq.(4.4)

(of the same class):

I

Q'(x) = q(x,t,Qo,6 o) - Q ' (x , t ) , Q(x) = Q(x,t,Qo,~o) - Q(x,t) (4.51)

which, of course, implies Q'(x,O) = Q(x,O) = Qo(X); then, the corresponding ref lec- t ion coeff ic ients R' (k , t ) and R(k, t ) , that evolve in time as

R' (k , t ) : exp [4ik 6~(-4k2)t] Ro(k) (4.52a1

R(k,t) : exp [4ik ~o(-4k2)t] Ro(k) (4.52b)

where Ro(k) = R(k,O) = R'(k,O), are related to each other by the transformation (4.5), R'(k) and R(k) being replaced in this equation by R' (k , t ) and R(k,t) given by (4.52a) and (4.52b) respectively. The transformation (4.5) is then consistent with the relat ionship which obtains by el iminating Ro(k) from (4.52), namely

R' (k , t ) = exp {4ik [6o(-4k 2) - Bo(-4k2)] t } R(k,t) (4.53)

i f we whoose

Therefore eqs.(4.53), (4.52) and (4.54) imply that, i f Q(x,t) sat is f ies the SNEE (3.1) with the i n i t i a l condition (4.6), then Q' (x , t ) which is related to Q(x,t) by the transformation

82

cos, ~,~ [~o<~ 0o<~>]~ [0 <x ~> 0<x< ~ s~n, ~'~ [~o<~ ~o<~>]~

I +° ~x<X~+0x<X~+I0<x~> ~<x,~>] ox [~<x<> 0<x~]~=o <~> JX

satisfies the SNEE

!

Q~(k) = 2B~(L') Qx(x,t) (4.56)

with the same i n i t i a l condition (4.6), Q'(x,0) = Qo(X); of course, the operator L'

is defined by (4.7) with Q(x,t) replaced by Q ' ( x , t ) .

For sake of completeness, we now show how to deal also with the translations

of the space variable x, within the present scheme based on the transformations

(4.4). We f i r s t , notice that, i f R(k) is the ref lect ion coeff ic ient corresponding

to the potential Q(x), then

R'(k) = exp(2ik AX) R(k) (4.57)

is the ref lect ion coef f ic ient corresponding to the potential

Q'(x) = Q(x + Ax) (4.58)

This result can be easily derived from the Schr~dinger equation (1.9) and from the

definit ion (1.10a) of the reflection coefficient. Comparing now (4.57) and (4.5)

shows that, to make (4.57) coincide with the transformation (4.5), we can choose

f (z 2) = cosh(z AX/2) , g(z 2) = -z I sinh(z Ax/2) (4.59)

and therefore the corresponding transformation (4.4) implies that Q(x+ AX) is rela-

ted to Q(x) by the operator identity (4.48) we have already derived (with f(x) re-

placed by Q(x) and a replaced by AX).

I t should be realized that more general transformations (some of them being re

ported by F. Calogero and A. Degasperis 127,111),can be obtained by combining two,

or more, of the four types of transformations we have discussed, namely the

B~cklund transformation, the resolvent formula, the transformation (4.55) and the

x-translation (4.48). As an example, we now write down the generalized resolvent

formula which relates Q(x+Ax, t+At) to Q(x,t), Q(x,t) being a solution of the

SNEE (3.1),

83

cosh{A1/2 [ (AX/2) + At Bo(A) ] } [Q(x+Ax, t + A t ) - Q(x , t ) ] -

A -1/2 s i nh {A 1/2 [(Ax/2) + At ~o(A)] } {ex(X + AX, t + At) + Qx(x,t) + %. J

Q(x+Ax, t+A t ) -Q(x , t ) ] dx' (X '+ t + At) - Q(x ' , t ) } : 0 (4.60) I JX

where, of course, A is the in tegro-d i f ferent ia l operator defined by (4.3) with Q'(x) = Q(x+Ax, t + A t ) and Q(x) = Q(x,t) . The interested reader w i l l f ind the de- r ivat ion of this equation in the reference 1271.

84

REFERENCES

1. Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for Solving the Korteweg-de Vries equation. Phys. Rev. Lett. I_99, 1095 (1967).

2. Ablowitz, MoJ., Kaup, D.J., Newell, A.C., Segur, H.: The Inverse Scattering transform-Fourier analysis for nonlinear problems. Appl. Math. 5_~3, 249 (1974); hereafter referred to as AKNS.

3. Scott, A.C., Chu, F.Y.F., McLaughlin, D.W.: The soli ton: a new concept in applied science. Proc. I.E.E.E. 61, 1443 (1973). Bollough, R.K.: Solitons. In In t~act ion of Radiation with Condensed Matter. Vol . l , IAEA-SMR- 20/51, Vienna (1977); also ' lectures delivered at the Interna- tional Advanced study Inst i tu te on Nonlinear Equations in Physics and Mathema- t ics , Istanbul, August 1-13, 1977. Proceedings edited by ~.0.Barut. ' . . . . .

4. Calogero, F., Degasperis, A.: Solution by the spectral transform method of a nonlinear evolution equation including as a special case the cyl indrical KdV equation. Lett. Nuovo Cimento 2__~3, 150 (1978).

5. Well known l inear problems are the Schr~dinger and the generalized (non selfad- jo in t ) Zackarov-Shabat spectral problems on the real l ine, with a potential which vanishes at i n f i n i t y . The most interesting associated evolution equations are the KdV for the f i r s t one and the MKdV (modified KdV), sine-Gordon and non- l inear Schr~dinger equations for the second one. A unified treatment of these evolution equations can be obtained by considering the N xN matrix Schr~dinger spectral problem: see Refo 1121 and Jaulent, M., Miodek, I . : Connection between Zackarov Shabat and Schr~dinger Type Inverse Scattering Transforms. Preprint PM/77/9, University of Montpell ier (1977). Calogero, F., Degasperis, A.: in pre paration. The SchrSdinger problem with a potential which depends in a simple way on the eigenvalue has been discussed by Jaulent, M. and Miodek, I . : Nonli- near evolution equations associated with "energy-dependent Schr~dinger poten- t ia ls " . Lett. Math. Phys, 1, 243 (1976). The Schr~dinger problem with a poten- t ia l which diverges at infTnity has been discussed by Kulish, P.: Inverse scattering problem for Schr~dinger equation on a l ine with potential growing in one direction. Mathematical Notes, Leningrad (1970). Also Calogero, F., Degaspe r is , A.: Inverse spectral problem for the one-dimensional Schr~dinger equation- with an additional l inear potential. Lett. Nuovo Cimento, 2__33, 143 (1978). See also Ref. 141.

6. Lax, P.D.: Integrals of Nonlinear equations of evolution and sol i tary waves. Comm. Pure Appl. Math. 2_~1, 467 (1968).

7. Calogero, F.: A Method to Generate Solvable Nonlinear Evolution Equations. Lettere Nuovo Cimento, 1__4, 443 (1975). Calogero, F., Degasperis, A.: Nonlinear Evolution Equations Solvable by the In- verse Spectral Transform. I. Nuovo Cimento 32B, 201 (1976).

Wahlquist, H.D., Estabrook, F.B.: Prolungation structures of nonlinear evolu- tion equations. J. Math. Phys. I_6_6, 1 (1975).

Corones, J. , Markovski, B.L., Rizov, V.A.: Bilocal Lie Groups and Solitons. Phys. Lett. 61A, 439 (1977).

Novikov, S.P.: New applications of algebraic geometry to nonlinear equations and inverse problems. To appear in the Proceedings of a Symposium held at the Accademia Nazionale dei Lincei in Rome in June 1977, Calogero, F. (editor): Nonlinear evolution equations Solvable by the spectral transform. Pitman. Lon- don (1978). ....

11. Calogero, F., Degasperis, A.: Nonlinear evolution equations solvable by the in- verse spectral transform.l l . Nuovo Cimento 39B, 1 (1977).

12. Wadati, M., Kamijo, T.: On the extension of inverse scattering method. Prog. Theor. Phys. 52, 397 (1974).

8.

9.

I0.

85

13. Newton, R.G.: Scattering Theory of Waves and Part ic les. Mc Graw Hi l l Book Com pany, New York (1966).

14. I t can be eas i ly proved that i f Q(x) is hermitian and k is real , then T(k) is not singular.

15. Gel'fand, I .M., Levitan, B.M.: On the determination of a d i f f e ren t i a l equa- t ion from i t s spectral function. Amer. Math. Soc. Transl.~ 1, 253 (1955). Agranovich, Z.S., Marchenko, V.A.: The Inverse Problem of Scattering Theory. ( translated from Russian by B.D. Seckler). NewYork, Gordon and Breach (1963). Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory. Springer Verlag, New York (1977).

16. For a detai led treatment ( in the case N = I ) see the paper by Calogero, F.: Generalized wronskian re lat ions, one-dimensional Schr~dinger equation and non- l inear par t ia l d i f f e ren t i a l equations solvable by the inverse-scatter ing me- thod. Nuovo Cimento 31B, 229 (1976).

17. Indeed th is resu l t applies to a l l SNEE's associated to the generalized Zacha- rov-Shabat spectral problem since they are a subclass of the present class of SNEE (see below and the references reported in footnote 151). Nonlinear evo- lu t ion equations which are not isospectral flows but can be s t i l l investigated by the ST method associated to the single-channel Schr~dinger problem and the generalized Zackarov-Shabat problem, have been discussed by Newell, A.: The general structure of integrable evolution equations, to appear in Proc. Roy. Soc., 1978; and by Calogero, F., Degasperis, A.: Extension of the spectral transform method for solving nonlinear evolution equations. I & I I . Lett Nuovo Cimento 2_22, 131 and 263 (1978).

18. Since the evolution equation (2.28) is invar iant under t ime-translat ions, there is no loss of general i ty in considering t=O as the i n i t i a l time.

19. Calogero, F., Degasperis, A.: Coupled nonlinear evolution equations solvable via the inverse spectral transform and sol i tons that come back: the boomeron. Lettere Nuovo Cimento 1_66, 425 (1976). Calogero, F., Degasperis, A.: B~cklund transformations, nonlinear superposi- t ion pr inc ip le , mul t iso l i ton solutions and conserved quant i t ies for the "boo- meron" nonlinear evolution equation. Lettere Nuovo Cimento 1__6, 434 (1976).

20. A special class of non hermitian potentials has been discussed by: Bruschi,M. Levi, D., Ragnisco, 0.: Evolution equations associated to the t r iangular ma- t r i x Schr~dinger problem solvable by the inverse spectral transform. Nuovo Cimento 45A, 225 (1978).

21. Scott:Russel, J. : Report on waves. Report of the Fourteenth Meeting on the Br i t i sh Association for the Advancement of Science, London, 1845, pp.311-390.

22. Zabusky, N.J. Kruskal, M.D.: Interact ion of sol i tons in a co l l i s ion less plas ma and the recurrence of i n i t i a l states. Phys. Rev. Lett. 15, 240 (1965) Zabusky, N.J.: A synergetic approach to problems of non l i n~ r dispersive wave propagation and interact ion. In Nonlinear Part ia l D i f fe ren t ia l Equations Ames, W. (ed i tor ) . New York: Academic Press (1967), pp.223-258.

23. Kruskal, M.D.: The b i r th of the sol i ton. Proceedings of the Symposium held at the Accademia Nazionale dei Lincei in Rome in June 1977. Calogero, F. (ed i tor ) : Nonlinear evolution equations solvabl e by the spec- t ra l transform. Pitman. London (1978).

24. Zackarov, V.E.: Kinetic equation for sol i tons. Soviet Phys. JETP, 33, 538 (1971). Wadati, M., Toda, M.: The exact N-soliton solut ion of the Korteweg-de Vries equation. J.Phys. Soc. Japan 32, 1403 (1972). Tanaka, S.: Publ. Res. Inst. Ma-'-th. Sci. Kyoto Universi ty 8, 419 (1972/73). See also: Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Korteweg- de Vries equation and general izat ion. VI. Methods for exact solut ion. Comm. Pure Appl. Math. 27, 97 (1974).

88

25. Condrey, P.J., Eilbeck, J.C., Gibbon, J.D.: The Sine-Gordon equation as a mo- del classical f ie ld theory. Nuovo Cimento 25B, 497 (1975).

26. In the special case ~A~ = 0 the nonlinear evolution equation (3.34a) coinci- des with the Landau,Lifshitz equation describing the magnetization of a ferro- magnetic substance.

27. Calogero, F., Degasperis, A.: Transformations between solutions of d i f ferent nonlinear evolution equations solvable via the same inverse spectral trans- form, generalized resolvent formulas and nonlinear operator ident i t ies. Lettere Nuovo Cimento I_6_6, 181 (1976).

28. See, for instance: Lamb Jr . , G.L.: B~cklund transformations for certain non- l inear evolution equations.J. Math. Phys. 15, 2157 (1974); Chen, H.H.: General derivation of B~cklund~ransformations from inverse scatte ring problems. Phys. Rev. Lett. 33, 925 (1974). Miura, R.M. (editor) B~cklund tran-sformations. Lectures Notes in Mathematics, 515, Berlin, Heidelberg, New York: Springer Verlag (1976). See also Ref. 1301.

29. Case, K.M., Chu, S.C.: Some remarks on the wronskian technique and the inverse scattering transform. J. Math. Phys. 18_8, 2044 (1977).

30. Wahlquist, H.D., Estabrook, F.B.: B~cklund transformations for solutions of the Korteweg-de Vries Equation. Phys. Rev. Lett. 3__11, 1386 (1973).

31. See, for instance: Kruskal, M.D.: Nonlinear wave equations, in Dynamical Sys- tems, Theory and Applications. Moser, J. (editor). Lectures Notes in Physics, 3__88, Berl in, Heidelberg, New York: Springer Verlag (1975).

87

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THE SPECTRAL TRANSFORM AS A TOOL FOR SOLVING NONLINEAR

DISCRETE EVOLUTION EQUATIONS

D. Levi

I s t i t u t o di Fisica - Universita di Roma, 00185 Roma - I t a l y

I s t i t u t o Nazionale di Fisica Nucleare, Sezione di Roma

Table of Contents

Page

1. Introduction 92

2. The Spectral Transform 92

3. Nonlinear Discrete Evolution Equations 96

4. Special Cases 99

5. Extension to nonlinear D i f f e ren t i a l - Difference Equations with N-dependent Coeff ic ients 101

References 104

92

1. INTRODUCTION

As i t is well known there is an increasing number of problems in natural

sciences which are described by nonl inear d i f f e r e n t i a l d i f ference equations.

Difference equations arise in many d i f f e ren t f i e lds of science, for instance

in the study of e lec t r i ca l networks, in s t a t i s t i c a l problems, in queueing problems,

in ecological problems, as computer models for d i f f e r e n t i a l equations, e tc . , that

is in a l l s i tuat ions in which some sequential re la t ion ex is t at various discrete

values of the independent var iable l l I . More s p e c i f i c a l l y , nonl inear discrete evol~

t ion equations have been studied as models for v ib ra t ion of par t ic les in an anharmo

nic l a t t i c e (the Fermi-Pasta-Ulam l a t t i c e 12I and the Toda l a t t i c e 131), for trea-

t ing ladder type e lec t r i c networks 141, as a model for wave exc i ta t ion in plasma

15t, etc.

We can solve the Cauchy problem for many l i near discrete evolut ion equations

by a discrete analogue of the Fourier transform 16I which shows i t s e l f an equal ly

powerful technique as the continuous one. In the las t decades i t has been in t rodu-

ced the Spectral Transform method as an extension of the Fourier technique to solve

the Cauchy problem for cer ta in classes of nonl inear evolut ion equations I71. In the

las t few years i t was recognized that the same kind of ideas, which are at the base

of the Spectral Transform method, were appl icable to the discrete case as well I81,

thus opening new perspectives in the study of nonl inear discrete evolut ion problems.

In Section 2 we shal l f i r s t review the passages necessary to solve l i near d is-

crete evolut ion equations by the discrete Fourier transform; then, s ta r t ing from

the Zakharov-Shabat d iscret ized eigenvalue problem, we shal l introduce the Spectral

Transform. In Section 3 the cor re la t ion between the evolut ion of the potent ia ls and

scat ter ing data w i l l be obtained through the Wronskian technique, obtaining at the

same time many propert ies of the discrete evolut ion equations, as, for example, the

B~cklund transformations. In Section 4 we shal l recover some of the important equa-

t ions belonging to th is class of nonl inear discrete evolut ion equations and in Sec-

t ion 5 we extend the method to equations with n-dependent coe f f i c ien ts , g iv ing, at

the end, the e x p l i c i t expression of the simplest new equation one obtains in th is

case.

2. THE SPECTRAL TRANSFORII

Let us consider a l i near d i f f e r e n t i a l - d i f f e r e n c e equation with constant coef-

f i c i en t s

u t (n , t ) = - i~(E+,E -) u (n , t ) (2.1)

03

where the funct ion u depends on the continuous var iable t 191 and on the discrete

var iable n. The s h i f t operators E ± are defined by

E±#(n) = # ( n ± l ) (2.2)

and the funct ion m is a real ent i re or rat ional funct ion of both i t s arguments.

Eq.(2.1) can be solved by applying the fo l lowing integral transform 161

b (z , t ) : ~ u(n , t ) z -n (2.3a)

u(n , t ) = ( i / 2~ i ) i b (z , t ) zn-ldz (2.3b)

where the loop integral in eq.(2.3b) is around the un i t c i r c le in the complex

z-plane, i f the i n i t i a l condit ions on u are such that b(z,0) is f i n i t e . We should

l i ke to stress that the integral transform (2.3) is jus t the analogue of the usual

Fourier transform and can be obtained by set t ing x=nA, z =exp(ikA) and re ta in ing

jus t the lowest term in A.

I .

The solut ion of the Cauchy problem for eq.(2.1) is obtained in 3 steps:

s tar t ing from the i n i t i a l data u(n,0) we calculate i t s transform in the z-space

b(z,0) = ~ u(n,0) z -n (2.4a)

I I . by applying the transformation (2.3) we f ind the l inear d i f f e ren t i a l equation

f u l f i l l e d by b ( z , t ) :

which y ie lds immediately:

b t ( z , t ) = - i ~ ( z , I / z ) b (z , t ) (2.4b)

b (z , t ) : b(z,0) e x p ( - i ~ ( z , I / z ) t ) (2.4c)

I I I . The solut ion of eq.(2.1) is obtained by applying formula (2.3b)

u(n,t) (1/2~i) ~ u(n ,0) zn-n'-lexp(-iw(z = ' , I / z ) t ) d z nl:=o~

(2.4d)

We would l i ke to point out that , as in the continuous case, the behaviour of the so-

l u t i on (2.4d) depends essent ia l l y on the dispersion funct ion m and thus i t is much

more transparent in the z-space than in conf igurat ion space.

94

The extension of the Fourier transform technique to t reat nonlinear d i f fe ren-

t i a l -d i f f e rence equations is obtained by introducing a Spectral Transform which, for

a given discrete spectral problem,establishes a one-to-one correspondance between

the potent ial u(n) and the spectral data. The f i r s t s who applied to the discrete

case the Spectral Transform technique have been Case and Kac 181, who introduced

the spectral transform for a d iscret ized version of the Schr~dinger eigenvalue pro-

blem

u(n) ~(n+l)+u(n-l) ~(n-l) = X~(n) (2.5)

where ~(n) is the eigenfunction and x the eigenvalue.

Here we shall instead tersely sketch the Spectral Transform method for a dis-

cret ized form of the Zakharov-Shabat eigenvalue problem 1101

~l (n+l) : ~l(n) z + Q(n) @2(n)

~2(n+i) = ~2(n)/z + R(n) ~l(n)

(2.6)

where @i(n) i = 1,2 are two f ie lds which take values only for discrete values of

the i r argument n, Q(n) and R(n) are two potent ials which go to zero as Inl go to in-

f i n i t y and z is a comples "eigenvalue". Eq.(2.6) is obtained from the Zakharov-Sha-

bat eigenvalue problem 1111

d~l(X)/dx : - i ~ l ( X ) + q(x) ~2(x)

d~2(x)/dx = i ~ 2 ( x ) + r(x) ~1(x)

(2.7)

by set t ing

= ( i /A) log z ; q(x) = Q(n)/A

x = nA ; r (x) : R(n)/A

d~i (x) /dx ~ (~i (n+l) - ~ i (n) ) /A

(2.8)

By introducing the vector ~_(n) of components ~l(n) and @2(n) eq.(2.6) reads

!(n+1) = (Z + Q(n) o++R(n) ~_) i ( n )

with

~± = (~I ± i~2)/2 ; Z = (I + ~3)z/2 + ( I - ~ 3 ) / 2 z

~j j = 1,2,3 being the Pauli matrices and I the iden t i t y of rank 2.

(2.9)

(2.10)

95

Start ing from the eigenvalue problem (2.9) for a given set Q(n), R(n) we can

define the spectral data

+ + +

S: {Z(k) ( IZ(k ) I > 1), C(k ) ; Z i k ) ( IZ i k ) l < 1), Cik ) , (k = 1, , N) ;

~+(z) ( Iz l ~ 1), g-(z) ( Iz l ~ 1)} (2.11)

through the asymptotic behaviour of the solut ion of eq.(2.9) , which, for the two i~

dependent vector solut ions of the scat ter ing process, reads:

I + zn ; B - (z )zn l { (n) = ( £ ( n ) , ~2(n)) ~ (2.12a)

n÷+ ~ (z)z-n; z-n /

l a+(z)z n ; 0 n l ~(n) - ~ (2.12b)

n ÷ - ~ k 0 ; ~-(z)z-

and for the N bound states

+ @±(n) r ~ C~k) X# sn - - n ÷+ ~ Z(k) (2.13a)

± ±n ~±(n) ~ ± x Z(k ) (2.13b) n ~ - ~

where × = , x- = and B+(z) (B-(z)) is the re f lec t ion coe f f i c ien t , func-

t ion of z ana ly t ic at least outside ( ins ide) the un i t c i rc le except for a set of N +

points Z(k ) (Z~k~)~, at which i t has a simple pole; i f the funct ions Q(n) and R(n) v~ nish faster than exponent ia l ly as Inl goes to i n f i n i t y , the constants C ± (k) are de- + f ined as the residues of ~±(z) at Z~k ).

From a given set of potent ia ls , q(n) and R(n), we obtain the spectral data S (2.11) through the d i rec t problem (2.9, 2.12, 2.13) and, s ta r t ing from the spectral

data S one recovers in a unique way the potent ia ls by solv ing the inverse problem through the Gelfand-Levitan-Marchenko discret ized matr ix equation:

K(n,n') + M(n+n') + i K(n,n") ~1 M(n'+n") = 0 n' >_ n , (2.14) n,,=n+1

where the kernel of the " in tegra l " equation (2.14) is defined by:

M(n) = (I - ~3 ) M+(n)/2 - ( I + ~3 ) M-(n)/2 (2.15)

96

N + M+-(n) : ( I /2= i ) B-+(z) zn-ldz ± k= "ZICTk) (Z~k))Tn-1, , (2.15)

and K(n,n') is a matrix of rank 2 whose diagonal elements give back the potentials

Q(n) and R(n)

Q(n) : - K22(n,n+1) ; R(n) = - K11(n,n+1) , (2.16)

Thus, i f Q~n) and R(n) depend on a parameter t , which we shall call time, also the

spectral data wi l l depend on t .

Consequently i f the evolution of Q(n) and R(n) is governed by a nonlinear dis

crete evolution equation, while the corresponding evolution of the spectral data is

simple, we have a way of f inding the solution of the Cauchy problem by an extension

of the procedure used in the l inear case in eqs. (2.4).

3. NONLINEAR DISCRETE EVOLUTION EQUATIONS

One way of obtaining the corresponding evolutions of the potentials and of the

spectral data is through the generalized Wronskian relations I121 which are obtai-

ned straighforwardly from the eigenvalue problem (2.9) 113]

P(n+Z)@'T(n) ~(m+(A) v+(n) + u_(A) v-(n)) @(n+l) = n=-=

~+(z) ; 1-P(-~)~'+(z)~-(z) ~ (z2)

~+(z2) +(z)# ' : (Z) ; B'-(z) / ~ l -P( -~)a+(z)~ ' - (z ) ; ~-(

I _ 0 -P(-~)~'+(z) a-(z) A+(z2)) + (3. I )

P(-~) a+(z) a-(z) A-(z 2) ; 0

R(n) , v-(n) = , where ~(n), respectively ~ ' (n) , are with v+(n) = \Q' (n) - e(n) / / "

two matrix solutions of (2.9) of the same "eigenvalue" z with potentials Q(n) and

R(n), respectively Q'(n) and R'(n), and scattering data S, respectively S'; ~'T(n) is the transpose of ~ ' (n) , P(n) is a scalar defined by the recursive relat ion

P(n) = P(n+l)(1-Q(n) R(n)) ; P(+=) = i

is an operator such that ±1 tion of x ,

x) is an arbi t rary polinomial func-

97

-l-co A±(x) = ~ {(~+ v_~(k), A((~±(A) - ~_+(x))/(A_- X)) v_-+(k-l))

(~_ v$(k), A((w_+(A_) - m±(x) ) / (A- x)) v-+(k ± 1)) }

where by ( , ) , here and in the fol lowing, we mean the scalar product, and A is a ma- t r i x operator of rank 2 such that

A ~/A(n) I = < A(n-1) + R' (n- l ) an(Sn(R(j) B(n) " Sn(Q'( j)) A(n))

B(n) B(n+l) + Q(n+l) a'n(Sn+z(R(j))B(n ) S ' - n+l(Q ( j ) ) A(n))

+ R(n) (S+(R'( j))B(n) - Sn(Q(j) ) A(n)) \ + , . ,) (3.2)

+ Q'(n)(Sn+l(R ( j ) )B(n) - Sn+I(Q(j))A(n))

with

a n = I - R(n) q(n) • a' = I - R'(n) Q'(n) ' n J a~ +

= Sn(y(j)) x(n) j !n (i=n ~ ~iil ) X(d)aj Y(J) S~(y(j)) x(n) = j=n ~ x(j ± 1) y ( j )

I f we set in eq.(3.1)

Q'(n) • Q(n) + Qt(n) dt

R'(n) ~ R(n) + Rt(n ) dt (3.3)

we obtain, at f i r s t order in dt, for w (x) = 1

( Inl I P(n+Z) ~T(n)

n =-~ \ Qt (n) /

I + ~(n+l) = -6t ; - B+(z)Bt(z) + P(-~) a+(z) at(z) ,

+

B (z) J (3.4)

F+(z) B-(z) + P(-~) ~+(z) a-(z) : i (3.5)

and for a generic ~(x)

P(n+l) ~T(n)~m(L) R(n) ~(n+l) = ~(z 2) ' (3.6) n=-~ -- ~Q(n) B+(z) 6-(z); ~-(z) /

where L is a matrix operator obtained from ~ b y keeping only the f i r s t order terms, i .e. by replacing everywhere in eq.(3.2) R' and Q' by R and Q. From eqs.(3.4, 3.6) i t follows that, i f Q(n) and R(n) evolve according to the nonlinear d i f fe ren t ia l -d i~

98

ference equation

/ I R t ( n ) ~ + w ( L ) S R ( n ) ) = 0 (3.7) ~ - Q t ( n ) / - \Q(n)

then the evolut ion of the scatter ing parameters of the spectral data is simple

given by

6~(z) ± m(z 2) 6±(z) = 0 (3.8a)

=~(z) : 0 (3.8b)

Eqs.(3.8) can be immediately solved to give:

6±(z,t) = B±(z,O) exp($m(z2)t) (3.9a)

~ ± ( z , t ) + = m-(z,O) (3.9b)

so that the transmission coe f f i c ien t is a constant of the motion and the re f lec t ion

coe f f i c ien t evolves simple. The evolut ion of the bound state parameters is simply

obtained by calculat ing the residue of 6±(z) at the poles

Z~k)(t) : Z~k)(O) (3.10a)

C~k)(t) : C~k)(O) exp(~m((Z~k))2)t ) (3.10b)

thus deducing that the flow is isospectral .

We should l i ke to stress that , as for the Fourier transform, the nonlinear

d i f f e ren t i a l -d i f f e rence equation (3.7) is characterized by the dispersion function

which determines the time evolut ion in the transformed space; i t should also be

noted the complete analogy between the Spectral Transform method for d i f f e r e n t i a l -

d i f ference equations and i ts continuous version 171: actual ly th is analogy guaran-

t ies the poss ib i l i t y of obtaining, also in the discrete case, B~cklund transforma-

t ions, the nonlinear superposition pr inc ip le , the general resolvent formula and an

i n f i n i t y of conserved quant i t ies. From this analogy we can also conclude that also

in the discrete case we have two kinds of solut ions, sol±ton solutions and back-

ground solutions and any general solut ion in coordinate space w i l l be a nonlinear

superposition of both. For the behaviour of these two kinds of solutions we refer

to reference 171 in this same issue.

In the fol lowing we jus t wr i te down the B~cklund transformations and, from one

of them, we shall derive the one sol±ton solut ion. Start ing from eq.(3.1) , i f the

fol lowing equation is f u l f i l l e d

99

~+(A__) i R(n) "~ + u_(A__)~R'(n)l:

Q'(n) / \Q(n) /

the r ight hand side of eq.(3.1) gives:

B'-+(z) =- (~±(z2) /~ (z2)) ~-+(z)

~'±(z) = m±(z2)/(m±(z 2) + a±(z2)) ~±(z)

0 (3.11)

(3.12a)

(3.12b)

showing thus that, i f Q(n) and R(n) is a solution of eq.(3.7) so is Q'(n) and R'(n).

Eq.(3.11) is thus a B~cklund transformation. Starting from the solution Q(n)=R(n)=O,

we are thus enabled to f ind the one sol i ton solution and, from i t , to build up the

"sol i ton ladder", that is , step by step, we can construct the N sol i ton solution.

In fact , sett ing ~+(x) = ax+b and u_(x) = c/x+d, we obtain from (3.11) the fo l lo -

wing nonlinear system of difference equations

Q'(n) = R ' ( n ) / ( R ' ( n + I ) R ' ( n - 1) ) - I / R ' ( n )

R'(n) = Q ' (n ) / (Q ' (n+ l ) Q ' (n -1 ) ) - 1/Q'(n) (3.13)

wich determine the one sol i ton solution as a function of n

÷ Q'(n) = C exp( - (p+(n+ l )+p_(~+ l ) ) ) sech(p_(n-~))

R'(n) = C-exp(p+n - p_~) sech(p_(n-~))

= (1/2p_) Iog(C + C-/(exp(p+ + p_) sh2p_))

p+ = log(z+z -) ; P_ = log(z+/z -)

(3.14)

while the time evolution is recovered by insert ing eqs.(3.14) into the evolution

equation and is given by eqs.(3.10).

4. SPECIAL CASES

in the following pages we want to show some of the ~nteresting equations one

can obtain from the class of nonlinear discrete evolution equations given by eq.

(3.7). We consider as a f i r s t example the equation one obtains i f we set R(n)=cQ*(n),

with ~ = ± 1, for Q(n) complex and w(x) = - i ( x - 2 + l / x ) , that is:

iQt(n) = (Q(n+l) + Q(n-Z)(Z+clQ(n)l 2) - 2q(n) (4.1)

100

This is a d i f f e r e n t i a l - d i f f e r e n c e vers ion of the non l inear Schr~dinger equation

1111, as by se t t i ng

q ( x , t ) = Q(n)/A

Q(n±l) = Aq ± A2qx

we get:

; X = nA ; T = A2t

+ (1/2) A3qxx ± (1/6) A4qxxx + O(A 5)

iqT = qxx + 2cq lql 2 + 0(A)

(4.2)

Qt(n) + (1 - ~Q2(n)) { Q(n- I ) - Q(n + i ) + ~Q(n)(Q2(N + 1) - Q2(n- 1))

+ Q(n - 2)(1 - eQ2(n - 1)) - Q(n + 2 ) ( 1 - ~Q2(n + 1) ) } = 0 (4.3)

which resu l t s as the appropr iate d iscre te equation fo r the modif ied Korteweg-de

Vries equat ion I141 as i t reduces to i t in the continuous l i m i t (4.2) together w i th

the d ispers ion r e l a t i o n and i t s s o l i t o n so lu t i on (up to terms of order A 2) 1151.

Another i n t e r e s t i n g equation obtains by choosing ~(x) = x - l / x :

Qt(n) + (1 -~Q2(n ) ) (Q(n-1) - q (n+ l ) ) = 0 (4.4)

This equation has been i n t e n s i v e l y studied as i t may represent a model f o r an e lec-

t r i c network as tha t given in f i g . ( 1 ) , w i th the inductance and the capaci tor depen-

ding non l i nea r l y on the cor rent and the po ten t ia l d i f fe rence :

L = arctan ( l ( n ) ) / l ( n ) ; C = arctan (V(n) ) /V(n)

eq. (4 .4) reads:

in f ac t , by se t t i ng (~ = -1) 1161

Q(mn) = -V(n) ; Q(2n-1) = - I (n )

dV(n) /d t = ( l + V 2 ( n ) ) ( l ( n + l ) - l ( n ) )

(4.5) d l ( n ) / d t = ( l + 1 2 ( n ) ) ( V ( n ) - V(n-1))

( fo r e = i i t corresponds to the model w i th L = t a n h - l ( l ( n ) ) / l ( n )

C = t a n h - l ( v ( n ) ) / V ( n ) which has been studied by Noguchi et a l . 1171). By the t rans-

format ion N(n) = (Q(n-1) + ~ ) ( Q ( n ) - ~-~)1161 eq. (4 .4) reduces to

A d i f f e r e n t class of equations obtains in the case R(n) = cQ(n) w i th Q(n) rea l .

By choosing ~(x) = x - 1 / x + x 2 - 1 / x 2 we obta in :

101

Nt(n) = N(n) (N(n+l) - N(n- l )) (4.6)

which represent a Volterra model 1181 of i n f i n i t e conf l ic t ing species; eq.(4.6) can

also describe the evolution of exci tat ions in plasmas 151. By set t ing 1161

u(n) = - log(N(2n) N(2n-Z))

for u(n) we get the Toda l a t t i ce equations 131

d2u(n)/dt 2 = 2e -u (n ) - e -u (n+ l ) - e -u(n-1) (4.7)

We have to notice that also from eq.(4.4) we can recover, in the continuous l i m i t ,

the modified Korteweg-de Vries equation, as, by defining 1121

Q(n) = Af(X,T) ; X = nA + 2At ; T = 3t/A 3

we obtain

fT = fXXX + 6~f2fx

5. EXTENSION TO NONLINEAR DIFFERENTIAL-DIFFERENCE EQUATIONS WITH N FDEPENDENT

COEFFICIENTS

As is well known, we can apply the Fourier transform method to solve the Cau-

chy problem for l inear discrete evolution equations with n-dependent coeff ic ients

at expenses of the s imp l i c i t y of the evolution equation in the transformed space.

In fact , choosing in eq.(2.1) ~ = (Wl(E +) + ~2(E-))n, eq.(2.4b) becomes:

b t ( z , t ) = - i (~ l (Z ) + ~2( i /z ) )z bz(z, t ) (5.1)

which is a part ia l d i f f e ren t ia l equation, whose Cauchy problem is solved by the me-

thod of character ist ics 1191.

I t has been shown 1201 that also the Spectral Transform method can be extended

to t reat nonlinear d i f fe ren t ia l -d i f fe rence equations wi~h n-dependent coeff ic ients

by introducing the new Wronskian re la t ion:

102

÷~

Z P(n+l ) @T(n) ~ ( ~ ) ( 2 n + l ) n=-m

2 ~(z )z( + \~2(z)6- (z)

p(-~)

where

{R(n) ~ @(n+l) : \q(n) /

;-(B+(z) Bz(Z) + P(-~)a+(z) mz(Z))~

+ B (z) / + P(-~) %(z) a-(z) ;-

; -P(-~) a+(z) a-(z) B+(z 2)

a+(z) a-(z) B-(z 2) ; 0 J v(z 2) : v+(z 2) + v_(z 2)

(5.2)

(5.3)

with u±(x) a polynomial function of i ts argument x ±1, and

+oo

B±(x) : ~ { (o+vv(k), L_((v+(L_) - v±(x))/(L_ - x)) (2k- 1) v (k - 1) k=-oo

(~_v(k), L_((v±(L_) - v±(x))/(L_- x)) (2k+Z) v ( k + l ) ) }

w h e r e v ( k ) = .

- Q ( k )

Eq.(3.7) thus reads:

(5.4)

/ R t ( n ) ~ w(L) /R (n )~ / R ( n l ) + _ + v(L)(2n+l) = 0

~-Qt(n ~xQ(n)// - ~Q(n (5.5)

and the corresponding l inear part ial d i f ferent ia l equations for the scattering data read:

B~(z) ± w(z 2) ~±(z) + v(z 2) z B~(z) = 0 (5.6a)

a~(z) + v(z 2) za~(z) ~ B$(z 2) a±(z) = 0 (5.6b)

which can be solved by the method of characterist ics 1191, thus showing that the flow is no more isospectral and the transmission coeff icients are no more constants of the motion.

Work is in progress on this new class of nonlinear d i f ferent ia l -d i f ference equations I15, 211; i t is worthwhile to show here just the simplest new equation one obtains in the case of R(n) = ~Q(n), for ~(z) = a (x -1 /x ) and v(x) =b (x -1 / x )

Qt(n) + (1- ~Q2(n))(Q(n-1)(2nb+a-b) - Q(n+ l ) (2nb+a+3b)) = 0 (5.7)

103

which by the t ransformat ion

N(n) = K(Q(n- 1) + ~-E-~)(Q(n) - / T ) (5.8)

can be reduced to the Vo l te r ra type equation

Nt(n ) = ~ N ( n ) ( N ( n - 1 ) ( 2 n - 3 + 6 ) - N ( n + l ) ( 2 n + 3 + 6 ) ) + ¥N(n) - 2~N2(n)

with ~, 6, ~ arbitrary non-zero constants, which is an equation v;ith a dissipative

term, a more rea l is t ic model for many physical phenomena 15, 221 .

L ( In - l )

I(In-1)

L ( I n ) L(In+1)

Vn-1 ,,I., Vn+1 T ,vn., i C(Vn,'q)

F i g . l . A ladder type non l inear LC c i r c u i t

104

REFERENCES

1. H. Levy and F. Lessman. F in i te Difference Equations, Macmillan, New York, 1961.

E. Pinney. Ordinary Di f ference-Di f ferent ia l Equations, Univ. of Cal i fornia Press, New York, 1958.

R. Bellman and K.L. Cooke. D i f fe ren t ia l -D i f fe rence Equations , Academic Press, New York, 1963.

G. Innis. Dynamical analysis in "Soft Science" studies; in defence of di f ference equations, in Lecture Notes in Biomathematics 2 (S. Levin ed.) , Springer Verlag, Berl in 1974, pp. 102-122.

2. E. Fermi, J.R. Pasta and S.M. Ulam. Studies of nonlinear problems, in Collected Papers of E. Fermi, Univ. of Chicago Press, Chicago, 1965, v o l . l l , pp.977-988.

N.J. Zabusky. A synergetic approach to problems of nonlinear dispersive waves propagation and in teract ion, in Proc. Symp. on Nonlinear Part ial D i f fe ren t ia l Equations (W.F. Ames, ed.) , Academic Press, New York, 1967, pp.223-258.

N.J. Zabusky and M.D. Kruskal. Interact ion of "Sol i tons" in a co l l i s ion less plas- ma and the recurrence of i n i t i a l states, Phys. Rev. Let t . I_.55, 240-243 (1965).

3. M. Toda. Vibrat ion of a chain with nonlinear in teract ion, J. Phys. Soc. Japan 2_~2, 431-436 (1967).

M. Toda. Wave propagation in anharmonic l a t t i c e , J. Phys. Soc. Japan 23, 501-506 (1967).

M. Toda. Mechanics and s ta t i s t i ca l mechanics of nonlinear chains, in Proc. of the Internat ional Conference on Stat is t iCal MechaniCs, K~oto 1968, Suppl. J. Phys. Soc. Japan 2_6_6, 235-2'3'7 (1969). . . . . . . . . . .

M. Toda. Waves in nonlinear l a t t i ce , Suppl. Progr. Theor. Phys. 45, 174-200 (1970).

M. Toda and 11. Wadati. A sol i ton and two sol i tons in an exponential l a t t i ce and related equations, J. Phys. Soc. Japan 34, 18-25 (1973).

R. Hirota. Exact l l -sol i ton solut ion of a nonlinear lumped network equation, J. ~n ~n Phys. Soc. Japan 3__55, Lo6-~o8 (1973).

M. H~non. Integrals of the Toda l a t t i ce , Phys. Rev. B_99, 1921-1923 (1974)

H. Flaschka. The Toda l a t t i ce I , existance of in tegra ls , Phys. Rev. B__99, 1924- 1925 (1974).

H. Flaschka. On the Toda l a t t i ce I I - Inverse scatter ing solut ion, Prog. Theor. Phys. 5__1_1, 703-716 (1974).

M. Kac and P. van Moerbeke. On an e x p l i c i t soluble systems of nonlinear d i f fe ren- t ia l equations related to certain Toda l a t t i ce , Advan. Math. 16, 160-169 (1975).

M. Kac and P. van Moerbeke. On some periodic Toda l a t t i ce , Proc. Nat. Acad. Sci. U.S.A. 72, 1627-1629 (1975).

M. Toda. Studies in a non-l inear l a t t i c e , Phys. Reports 18C, 1-125 (1975).

E. Date and S. Tanaka. Analogue of the inverse scatter ing theory for the discrete H i l l ' s equation and exact solutions for the periodic Toda l a t t i ce , Prog. Theor. Phys. 55, 457-465 (1976).

M. Toda. Development of the theory of a nonlinear l a t t i ce , Suppl. Progr. Theor. Phys. 59, 1-35 (1976).

R. Hirota and J. Satsuma. A var ie ty of nonlinear network equations generated from the B~cklund transformation for the Toda l a t t i c e , Suppl. Prog. Theor. Phys. 59, 64-100 (1976).

105

K. Sawada and T. Kotera. Toda la t t i ce as an integrable system and the uniqueness of Toda's potent ia l , Suppl. Prog. Theor. Phys. 59, 101-106 (1976).

M. Toda, R. Hirota and J. Satsuma. Chopping phenomena of a nonlinear system, Suppl. Prog. Theor. Phys. 59, 148-161 (1976).

H. Flaschka and D.W. McLaughlin. Canonically conjugate variables for the Korte- weg-deVries equation and the Toda l a t t i ce with periodic boundary condit ions, Prog. Theor. Phys. 55, 438-456 (1976).

B.A.Dubrovin, V.B. Matveev and S.P. Novikov. Nonlinear equations of Korteweg-de Vries type, f in i te-zone l inear operators, and abelian var ie t ies , Russian Math. Surveys 31, 59-146 (1976).

R. Hirota. Nonlinear par t ia l di f ference equations I I - Discrete Toda equation, J. Phys. Soc. Japan 43, 2074-2078 (1977). R.K. Dodd. Generalize-a~B~cklund transformation for some nonlinear par t ia l d i f f e - rence equations, J. Phys. A l l , 81-91 (1978).

4. R. Hirota and K. Suzuki. Studies on l a t t i ce sol i tons by using e lec t r i c networks, J. Phys. Soc. Japan 2_88, 1336-1337 (1970).

R. Hirota. Exact N-sol i ton solut ion of nonlinear lumped self-dual network equa- t ions, J. Phys. Soc. Japan 3-5, 289-294 (1973).

5. V.E. Zakharov, S.L. Musher and A.M. Rubenchik. Nonlinear stage of parametric wave exc i ta t ion in a plasma, Sov. Phys. J.E.P.T. Lett . 19, 151-152 (1974).

6. E.C. Titchmarsh. Theory of Fourier in tegra ls , Oxford Univ. Press Oxford 1937, pp.298-301.

E.C. Titchmarsh. Solutions of some functional equations, J. London Math. Soc. 14, 118-124 (1939).

7. F. Calogero. Spectral Transform and nonlinear evolut ion equations, on this same issue, pp. 29.

A. Degasperis. Spectral Transform and so lvab i l i t y of nonlinear evolut ion equa- t ions, on this same issue, pp. 35.

8. K.M. Case and M. Kac. A discrete version of the Inverse Scattering problem. J. Math. Phys. 14, 594-603 (1973).

K.M. Case. On discrete Inverse Scattering problem I I , J. Math. Phys. 14, 916- 920 (1973).

K.M. Case and S.C. Chiu. The discrete version of the Marchenko equations in the Inverse Scattering problem. J. Math. Phys. 14, 1643-1647 (1973).

K.M. Case. The discrete Inverse Scattering problem in one dimension, J. Math. Phys. 1-5, 143-146 (1974).

K.M. Case. Fredholm determinants and mult ip le sol i tons, J. Math. Phys. 17, 1703- 1706 (1976).

9. Here, and allways in the fo l lowing, subscripts indicate par t ia l d i f f e ren t ia t i on : u t = ~u/~t, etc.

10. M.J. Ablowitz and J.F. Ladik. Nonlinear d i f f e ren t ia l -d i f f e rence equations, J. Math. Phys. 1-5, 598-603 (1975).

M.J. Ablowitz and J.F. Ladik. Nonlinear d i f f e ren t ia l -d i f f e rence equations and Fourier analysis, J. Math. Phys. 17, 1011-1018 (1976).

M.J. Ablowitz and J.F. Ladik. A nonlinear di f ference scheme and Inverse Scatte- r ing, Stud. Appl. Math. 5-5, 213-229 (1976).

M.J. Ablowitz. Nonlinear evolution equations-continuous and discrete, Siam Review 1-9, 663-684 (1977).

M.J. Ablowitz. Lectures on the Inverse Scattering transform, Stud. Appl. Math. 58, 17-94 (1978).

106

11. V. Zakharov and A.B. Shabat. Exact theory of two-dimensional sel f- focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet. Phys. J.E.P.T. 3_44, 62-69 (1972).

12. S.C. Chiu and J.F. Ladik. Generating exact ly soluble nonlinear discrete evolu- t ion equations by a generalized Wronskian technique, J. Math. Phys. 1__8, 690-700 (1977).

13. For another method of obtaining the nonlinear discrete evolution equations, see the works of re f . 8

14. M. Wadati. The exact solut ion of the Modified Korteweg-deVries equation, J.Phys. Soc. Japan 3_22, 1681 (1972).

15. M. Bruschi, D. Levi and O. Ragniscoo Spectral Transform approach to a discrete version of the Modified Korteweg-de Vries equation with x-dependent c o e f f i c i e n t , sent to the I I Nuovo Cimento for pubblication.

16. M. Wadati. Transformation theories for nonlinear discrete systems, Suppl. Prog. Theor. Phys. 59, 36-63 (1976).

17. A. Noguchi, H. Watanabe and K. Sakai. Shock waves and hole type sol i tons of non- l inear selfdual network equation, J. Phys. Soc. Japan 43, 1441-1446 (1977).

18. V. Volterra. Lecons sur la th~orie math~matique de la lu t te pour la v ie , Gau- t i e r - V i l l a r s , Paris 1931.

R. Hirota and J. Satsuma. N-solitons solutions of nonlinear network equations describing a Volterra system, J. Phys. Soc. Japan 40, 891-900 (1976).

S. Fu j i i , F. Kako and N. Mugibayashi. Inverse method applied to the solut ion of nonlinear network equations describing a Volterra system, J. Phys. Soc. Japan 42, 335-340 (1977).

J.F. Ladik and S.C. Chiu. Solution of nonlinear network equations by the Inverse Scattering method, J. Math. Phys. I__88, 701-704 (1977).

M. Wadati and M. Watanabe. Conservation laws of a Volterra system and nonlinear self-dual network equations, Prog. Theor. Phys. 57, 808-811 (1977).

19. G.F. Carr ier and C.E. Pearson. Part ia l d i f f e ren t i a l equations~ theory and tech- 9~que, Academic Press, New York 1976.

20. D. Levi and O. Ragnisco. Extension of the Spectral Transform method for solving nonlinear d i f f e ren t ia l -d i f f e rence equations, Lett . Nuovo Cimento (1978), 22 691-696.

21. M. Bruschi, D. Levi and O. Ragnisco. Spectral Transform approach to a discrete version of the nonlinear Schr~dinger equation with x-dependent coef f ic ients , in preparation.

22. J.C. Fernandez and G. Reinisch. Collapse of a Volterra sol i ton into a weak mono- tone shock wave, Physica A (1978) in press.

HYPERBOLIC BALANCE LAWS IN CONTINUUM PHYSICS

C.M. Dafermos

Brown Univers i ty , Providence RI, USA

Table of Contents

1. Introduct ion

2. Kinematics of Bodies

3. Balance Laws

4. Const i tu t ive Relations

5. Hypere last ic i ty

6. Hyperbolic Balance Laws

7. Shock Waves

8. Entropy

References

Page

108

108

110

113

113

115

116

117

121

108

1. INTRODUCTION

Af ter ha l f a century of decl ine, there has been, in the las t three decades, a

revival of in te res t in continuum physics, due par t l y to the development of new mate

r i a l s of technological importance (high polymers, rubbers, l i qu id crysta ls) and pa L

t l y to the rea l i za t ion that modern Geometry and Analysis can be employed in order

to recast the classical theories into a more rat ional formulat ion and solve old pro

blems that had been abandoned as unsolvable. In the f i r s t f i ve sections of these

lecture notes, I attempt to present a b i rd 's-eye view of the st ructure of continuum

physical theories today°

In rough terms, a continuum physical theory consists of a co l lec t ion of balance

laws and a set of cons t i t u t i ve re la t ions . The former i den t i f i e s the framework of

the theory (e .g . , continuum mechanics, continuum thermomechanics, continuum e lec t ro

dynamics, e tc . ) whi le the l a t t e r characterizes the nature of the continuous medium

(e.g. , e las t i c so l id , viscous f l u i d , e tc . ) . In Section 3 we discuss the general

form of a balance law and wr i te down, as an i l l u s t r a t i o n , the balance laws of cont~

nuum thermomechanics.

The major success of modern continuum physics is the development of a rat ional

theory of cons t i t u t i ve re la t ions , founded on the pr inc ip les of frame ind i f ference,

material symmetry and an appropriate in te rp re ta t ion of the second law of thermodyna

mics. No attempt w i l l be made here to present th is theory in i t s genera l i ty ; howe-

ver, the basic ideas w i l l be discussed in the framework of the very simple concrete

example of hypere las t i c i t y (Section 5). In the remainder of the lecture notes, we

study certain aspects of the theory of nonl inear hyperbolic systems that resu l t

from the balance laws of continuum physics. This is a branch of the theory of par-

t i a l d i f f e r e n t i a l equations that is being studied vigorously at the present time°

We discuss here the main d i f f i c u l t i e s of the subject, namely nonexistence of smooth

solut ions (breaking of waves, development of shock waves) and nonuniqueness of weak

solut ions and we show how ideas o r ig ina t ing in continuum physics (the second law of

thermodynamics) can be employed in order to s ingle out the phys ica l ly admissible

so lu t ion.

I wish to thank the organizers of the symposium and in par t i cu la r Antonio F.Ra

~ada and Luis and Luciana Vazquez for t he i r hosp i t a l i t y .

2. KINEMATICS OF BODIES

In continuum physics, the mathematical model of a bod~ is a manifold characte-

r ized by a reference conf igurat ion, that is an open subset B of the reference space

R 3. The typ ica l point X G B w i l l be cal led a mater ia l part icu le ,. In the appl icat ions,

109

the standard method of construct ing a reference conf igurat ion of a moving body is

to i den t i f y "molecules" with the point in space that they happen to occupy at a cer

ta in f ixed time instant . In general, however, a reference conf igurat ion need not be

an actual conf igurat ion of the body but only an abstract three-dimensional "picturg'

of i t . As a matter of fact i t is convenient to assume throughout that the reference

space is not necessari ly the physical space but jus t a copy of i t .

A conf igurat ion of the body B is a diffeomorphism x = x(X) from B to the phy-

s ical space R 3. A motion of B is a family x = x(X, t ) of conf igurat ions, parametri-

zed by time t . Thus for f ixed (X , t ) , x(X, t ) w i l l be the posi t ion of par t i c le X at

time t ; for f ixed t , the map x ( . , t ) : B ÷ R 3 is the conf igurat ion of the body at

time t ; f i n a l l y , for f ixed ~X 6 B, the curve x(X, . ) is the t ra jec tory of pa r t i c le X.

In continuum physics one seeks to determine the evolut ion in time of various

f ie lds of physical quant i t ies , such as temperature, e lec t r i c f i e l d , etc. defined

over the moving body. Since every conf igurat ion is a diffeomorphism of the refe-

rence conf igurat ion, the f i e l d quant i t ies can be represented equally well as func-

t ions of (X, t ) ( re fe ren t ia l descr ip t ion)or ( x , t ) (spat ial descr ip t ion) . Both repre-

sentations are useful . This induces a nasty notat ion problem since three d i f f e ren t

symbols are needed for each f i e l d quant i ty , namely one for the value of the quant i-

ty , one for i t s re ferent ia l descr ipt ion and one for i t s spat ial descr ipt ion. For

example, in the case of temperature,

@ = ~(X,t) = #(~, t ) (2.1)

In order to circumvent th is cumbersome notat ion one general ly i den t i f i es funct ions

with the i r values, using, for example, @ to denote the value of temperature as

well as the functions ~ and # in (2.1). To avoid confusion in the representation

of par t ia l der ivat ives one makes the fo l lowing convention: Cartesian components of

~, in reference space, w i l l be designated with Greek superscr ipts, such as X ~, X B,

X ¥, e tc . , whi le Cartesian components of ~, in physical space, w i l l be designated

with lower case Latin superscr ipts, e.g. , x i , x j , x k, etc. In the re fe ren t ia l repre

sentation of a f i e l d , the par t ia l der ivat ives ~ and ~ t w i l l be denoted by ,~ ~X ~

and a dot ( . ) , respect ive ly , whi le in the spat ia l representation 2. and -~t w i l l ~x 1

by , i and ~-~ . Thus, in the example (2.1) of temperature, be denoted

~ - 8 ~ - 0 ' ~# = @ i ~ t ~ t ~X m ,~ ' ~t ~x I , '

We also adopt throughout the usual repeated index summation convention. I t turns

out that the above notat ion is very e f f i c i e n t and the reader rap id ly becomes fami-

110

l i a r with i t .

The basic kinematic f ie lds associated with a motion ~ = ~(~, t ! of the body,

are v.eloc.ity ~ = ~ and deformation gradient F = v x ~ (components F ]~ = x la) . Since

every conf igura t ion is a diffeomorphism of the re fe rence conf igura t ion we must have

det F ~ O.

Chain rule induces obvious re lat ions between the various part ia l der ivat ives.

For example, in the case of temperature,

28 vi@,i F i ( ~ = ~ + ,(~ = 8 . ,0¢ O~ , I

3. BALANCE LAWS

The re ferent ia l descr ipt ion of a balance law for a moving body B is an equa-

t ion of the form

(3.1)

that holds for every smooth subset ~ of B with boundary ~Q. Equation (3.1) states

that the quanti ty whose density is G is conserved in the sense that the time rate

of change of the amount contained in ~ is balanced by the rate of f lux through ~

and the rate of production in ~. In (3.1) G and H are f i e lds , i . e . , G = G(X,t),

H = H(~,t) . However, P does not only depend on (X,t) but also upon character is t ics

of a~ in the v i c i n i t y of X. The crucial idea was contributed by Cauchy who postula-

ted ( for the typical case of balance of momentum) that P may depend solely on the

or ientat ion of ~ at X, that is , P = P(X,t,N) where N is the uni t normal on ~ at

the point X. Cauchy then proceeded to prove that the dependence upon ~N is necessa-

r i l y l inear , i . e . ,

P = P ~ ( X , t ) N (3.2)

The reason of th is surpr is ing property is that , as ~ shrinks about a point, the vo-

lume of ~ goes to zero faster than the area of ~ and, as a resu l t , the f lux term

I PdS must balance loca l ly in i t s e l f . For a formal proof, using the celebrated

"Cauchy tetrahedron argument" see, e.g. 111.

In view of (3.2), (3.1) takes the form

d-~-I G dX~ = I P~N~dS + I H dx dt ~ (3.3)

111

Applying the Gauss-Green theorem, one easily shows that (3.3) is equivalent to the

f ie ld equation

= P~ + H (3.4)

The cumbersome integral equation (3.1) has thus been replaced by a d i f ferent ia l

equation. I t is this event that makes continuum physics workable~

The balance law (3.3) also admits the spatial description

I gdx = i pinids + I hdx (3.5) dt )m(t) ~ ~m(t) m(t)

where m(t) is the configuration of ~ at time t , n is the unit normal on ~w(t) and

g (det F)-IG h = (det F)-III , pi=(det - i i , F) FaR (3.6)

Since ~(t) varies with t ,

d ( f ~ d-t-]m(t)gdx = J m ( t ) " ~ d x + ~mlt)gvlnids

so that (3.5) yields the f ie ld equation

~g + (gv i) i + h (3.7) ~t , i = P,i

Continuum mechanics is characterized by the balance laws of mass, l inear momen

rum and moment of momentum. For the referent ial description of the balance law of

mass, one takes the reference mass density 9 o for G and zero for P~ and H. Thus

(3.4) yields ~o = O, i . e . , the reference density is independent of t . For the spa-

t ia l description of the some balance law, one takes the spatial mass density p (det 4) -1 = 9 ° for g and zero for pi and h, so that (3.7) gives

~p ~ + (~v l ) , i = 0 (3.8)

In the referential description of the balance law of l inear momentum G is Po~' P~

the Piola-Kirchhoff stress tensor, and H is Po~' the body force. Thus the is T i , f ie ld equation is

9o ~i = T~ + Pobi (3.9)

In the corresponding spatial description, g is p~, pi is T], the Cauchy stress ten-

112

sor, and h is pb. The f i e l d equation reduced with the help of (3.8), takes the form

PGi = Tj i , j + Pbi (3.10)

The f i e ld equations of the balance law of moment of momentum can be reduced consi-

derably with the help of the previous balance laws and end up in algebraic form:

F i T ~, : F j T~, (3.11)

T! = "~J (3.12) j I

Thus the Cauchy stress tensor ~s symmetric.

In continuum thermomechanics, the balance law of energy should be added to the

above l i s t . I ts re ferent ia l descript ion is obtained by insert ing po E + ½ Po I~ 12

for G, T~ v i + Q~ for P~ and P^bivi + for H, where is the internal energy, 1 u POr is the heat f lux vector and r is the energy supply. For the spatial descript ion we

take p~ + ½ pl~l 2 for g, T~v j + qi for pi and PbiVi + pr for h. The corresponding

f i e l d equations, reduced with the help of the balance laws of mass and momentum,

are

" = T~v i + Q~ + Dot (3.13) PO ~ i ,~ ~ '

j i ~ + pr (3.14) p~ = T iv , j + q j

The above l i s t of balance laws should be supplemented with the second law of ther-

modynamics, expressed by the Clausius-Duhem inequal i ty

0~. r > Pon - ( ),~ - Po g-- 0 , (3.15)

i r > p~ - ( n~6-),i~ - p ~ _ 0 , (3.16)

in re ferent ia l and spat ial descr ipt ion, respect ively, where n is the specif entropy

and g is the absolute temperature.

In continuum electrodynamics we also have the balance laws of e lec t r i c charge

and magnetic f l ux but we shall not wr i te them down since the examples considered

thus far are probably su f f i c i en t form to i l luminate the structure of balance laws

in continuum physics.

113

4. CONSTITUTIVE RELATIONS

Const i tut ive re lat ions relate the various f ie lds that enter in the balance

laws and characterize the nature of the continuous medium (e.g. viscous f l u i d , plas

ma, e las t ic d i e l e c t r i c , e tc . ) . For example, cons t i tu t i ve re lat ions determine, in

continuum mechanics, stress from the kinematical quant i t ies and, in continuum ther-

momechanics, stress, in ternal energy, heat f l u x and entropy (or temperature) from

the kinematical quant i t i tes and temperature (or entropy).

Const i tu t ive re lat ions must be compatible with the balance law of moment of

momentum, (3.11), (3.12), as well as the Clausius-Duhem inequa l i t y (3.15), (3.16).

They must also comply with the pr inc ip le of frame indi f ference which relates the

f ie lds corresponding to any two motions d i f f e r i ng by a r i g i d ro ta t ion . F ina l l y ,

cons t i tu t i ve re lat ions must r e f l ec t the type of material symmetry (e.g. isotropy)

with which the medium happens to be endowed.

In the fo l lowing section we show, in the framework of a simple example, how

cons t i tu t i ve re la t ions are reduced in order to comply with the above requirements.

There is an extensive theory of cons t i tu t i ve re lat ions for which the reader is re-

ferred to 121.

5. HYPERELASTIClTY

In the framework of continuum mechanics, a material is hyperelast ic i f the

P io la-Ki rchhof f stress is determined from deformation gradient by a cons t i tu t i ve re

la t ion of the form

T~ = ~W(~). (5.1) 1 ~F 1

The s t ra in energy funct ion W(~) is interpreted as the density of the (potent ia l ) m R

chanical energy stored in the body by deformation.

In the context of hypere las t i c i t y , the pr inc ip le of frame indi f ference states,

that s t ra in energy should be unaffected by r i g id ro ta t ions, that is the s t ra in ener-

gies of any two conf igurat ions x = x(X) and x = x*(X) = Ox(X), wi th ~oTo~ = I , are the

F* * same. Since ~. = VxX = 0 VxX = ~OF' the above requirement means

W(~) = W(O~), for a l l ~ nonsingular, 0 orthogonal (5.2)

By the polar decomposition theorem we may wr i te F = ~ , where the ro ta t ion tensor

R is orthogonal and the r i gh t stretch tensor U = JFTF is %nnmetric and pos i t ive r ~ r ~ r u

de f in i t e . Applying (5.2) with 0 = R T we deduce

114

w = w(~) , ( 5 . 3 )

that i s , frame indif ference precludes dependence of s t ra in energy upon the rotat ion

tensor and allows only dependence on the r igh t stretch tensor. I t is more convenient

to v isual ize W as a function of the r igh t Cauchy-Green stra in tensor ..... C=U2=FTF,~

i . e . ,

w = w(c) ( 5 . 4 )

I t is easy to check that the Pio la-Kirchhof f f stress, determined by (5.1),with

W given by (5.4), sa t i s f ies automatical ly the balance law of moment of momentum

(3.11). This happy coincidence, however, is pecul iar to hypere las t ic i ty since in

more general theories the balance law of moment of momentum imposes on const i tu t ive

re lat ions addit ional res t r i c t ions to those dictated by frame indi f ference.

We now consider material symmetry in hypere las t ic i ty . I f W(~) and W*(~*) are

the st ra in energy functions of the same material re la t i ve to two d i f fe ren t reference

configurations B and B*, related via the diffeomorphism X = ~(X*), we must have

W(~) = W*(~*) , whenever ~C* = ~ ~,HTcH (5.5)

where H = ~X/~X*. When the material is endowed with symmetry, certain reference con tls ru

f igurat ions B* w i l l be indist inguishable from B in the sense that W*(.) = W(.). In

order to i den t i f y those B* and on account of (5.5) we consider the class r of unimo

dular matrices ~H (we need Idet H I = i to avoid changes in reference density) with

the property

W(~) = w(HT~H) , for a l l symmetric posi t ive def in i te ~ . (5.6)

I t is eas i ly ve r i f i ed that F forms a subgroup of the unimodular (or special l inear)

group SL 3. Thus r is cal led the isotropy group of the mater ia l , re la t i ve to the re-

ference configuration B. In practice one observes the material symmetry of a given

material determines the isotropy group ~ and then seeks the form of the st ra in ene~

gy function W(~) which is compatible with that r. In order to i l l u s t r a t e th is pro-

cedure we w i l l now discuss two examples.

A hyperelast ic material with maximal symmetry, i . e . , r = SL 3, is called a

hyperelast ic f l u i d . In th is case (5.6) w i l l be sa t is f ied for every unimodular matrix

so we may select , in par t i cu la r , H = HT = (det ~)1/6 ~ -1 /2 in which case

H T ~ = (met ~) i /3 ~. Since det ~ = (det F)2 = p~/p2 we deduce that a hyperelastic

material is a f l u i d i f and only i f the s t ra in energy is a function of density,

115

W = W(p). A simple calculation yields

~ : P--~2 dW(p) a! (5.7) J - PO dp J '

that i s , the Cauchy stress in a hyperelastic f l u id is a hydrostatic pressure depen-

ding on density. Thus the hyperelastic f l u id is just the ideal f l u id of classical

physics.

As a second example we consider the isotropic hyperelastic so l id , characteri-

zed by the condition that the isotropy group r is the f u l l orthogonal group. Thus

properties of the material are the same in every di rect ion in reference space. To

determine the form os st ra in energy we diagonalize the symmetric matrix ~,

OLO T, where oT = ~ ~ = I , and then inser t ~ = ~ in (5.6). I t follows that W = W(~)

is a syn~netric function of the eigenvalues of the Cauchy-Green stra in tensor ~. Equ~

va lent ly , one may v isual ize the st ra in energy as a function W = W(I, I I , I I I ) of

invar iants I = t r ~, I I = - ~ ( t r C)2 _ t r ~2 , I I I = det C of the pr incipal

A typical problem in hypere last ic i ty theory is to determine the motion

= ~(~, t ) of a body B by solving the f i e l d equations (3.9) for given reference de~

s i t y po(~) and body force ~(~, t ) and with prescribed i n i t i a l conditions ~(X,O),

~(~,0) and boundary conditions (assigning, for example, forces on the boundary).

This is a formidable problem and very l i t t l e is known concerning even gross qual i -

ta t i ve behaviour, such as existence, uniqueness and s t a b i l i t y of solut ions. The di~

cussion of hyperbolic systems in the fol lowing sections w i l l shed some l i gh t on the

d i f f i c u l t i e s of the problem.

6. HYPERBOLIC BALANCE LAWS

In the remainder of the notes we shall be looking into the theory of balance

laws from a somewhat more formal is t ic point of view. We ca l lec t a l l balance laws

of our theory into a single vector-valued balance law

: P~ + H" ( 6 . 1 ) V ~5~

where ~, ~P~ and ~ are n-dimensional vector f ie lds whoseoCartesian components w i l l

be denoted by cap i t a l Latin s u b s c r i p t s , such as V A, PX, H B, e t c . The s t a t e of each

par t ic le w i l l be expressed by an n-dimensional state vector U. The f ie lds V, ~P~ and

H wi l l be determined from U through smooth c o n s t i t u t i v e type r e l a t i o n s % %

: : : t ) ( 6 . 2 )

116

(as before, we iden t i f y functions with the i r values). We assume that U ÷ V(U) is a

diffeomorphism so that (6.1) becomes a system of equations for determining ~ ( ~ , t ) .

For a homogeneous (Po = const.) hyperelast ic material the state vector = (F~, "~ _ .3 _ 1 P? . . . . . r3' v , v 2, v 3,~ is 12-dimensional and the l i s t of balance laws

reads ~i = v i i ,~ = 1,2,3

OoV i" = T#I,~ + Pobi i = 1,2,3.

We may rewri te (6.1) in the form

(6.3)

~VA ~PA ~U B UB - ~U B UB,~ + HA (6.4)

System (6.4) w i l l be cal led hyperbolic i f for any f ixed UGR n and every 3-dimensio-

nal uni t vector~ the eigenvalue problem

( x v ~ + C vP ~) E = 0 (6.5)

has real eigenvalues and n l i nea r l y independent eigenvectors. The physical in terpre

ta t ion of hyperbol ic i ty is understood when one considers weak waves, that is solu-

t ions of (6.4) which are cm-smooth, m ~ O, but whose der ivat ives of order m+ l expe

rience jump d iscont inu i t ies across a propagating surface (wave). One easi ly shows

that i f the di rect ion of propagation is ~, then the speed of propagation and ampli-

tude of the wave are, respect ively, an eigenvalue and a corresponding eigenvector

of (6.5).

In par t i cu la r , (6.3) is hyperbolic i f and only i f the st ra in energy function

W(~) sa t i s f ies the Hadamard (or strong e l l i p t i c i t y , or rank-one convexity) condi-

t ion

~2w(~) ~]~J~ ~ B > 0 (6.6)

F ~ ~F J B g

for a l l nonzero 3-vectors ~ and ~.

7. SHOCK WAVES

From the point of view of analysis, the d i f f i c u l t y with nonlinear hyperbolic

systems is that in general there are no g lobal ly defined smooth solut ions. This is

due to the property that wave speeds (the eigenvalues of (6.5)) are not constant

but depend on the solut ion i t s e l f so that waves catch up with one another, are am-

p l i f i e d , and eventual ly break. One is then looking for solutions in the class of

117

functions that are smooth except on a fami ly of smooth propagating surfaces (shock

waves) across which they experience jump d i scon t inu i t i es . A funct ion U(X,t) in th is

class is a weak solut ion of (6.1) ( that is a solut ion of the balance law in in te -

gral form) i f i t sa t i s f ies (6.1) at every point of smoothness whi le across surfaces

of d i scon t inu i t y the Rankine-Hu~oniot jump condit ions

s - ( ; . 1 )

hold, where ~ is the d i rec t ion and s is the speed of propagation of the wave and a

bracket denotes the jump of the enclosed quant i ty across the surface of d iscont i -

nu i ty .

The class of piecewise smooth functions is not s u f f i c i e n t l y broad to encompass

the solut ions of (6.1) under a l l (even C~-smooth~) i n i t i a l data. I t has been conje~

tured that solut ions of (6.1) are gener ica l ly piecewise smooth but so far th is has

only been established in the case of a very simple model equation. I t seems that so

lu t ions of (6.1) should be sought in the class of funct ions of bounded var ia t ion in

the sense of Tonel l i -Cesar i , i . e . , the class of l oca l l y bounded measurable func-

t ions whose d i s t r i bu t i ona l der ivat ives are l oca l l y Borel measures. Functions of

bounded var ia t ion are endowed with a geometric structure that resembles the st ruc-

ture of piecewise smooth functions so that , in the framework of these solut ions,

one may ta lk , in an appropr iately generalized sense, about shock waves, the Rankin~

Hugoniot condi t ions, etc.

8. ENTROPY

One of the remarkable features of the theory of discontinuous solut ions of

(6.1) is nonuniqueness. In continuum thermomechanics, in order to single out the

phys ica l ly admissible so lut ions, one makes an appeal to the Clausius-Duhem inequa-

l i t y (3.15), (3.16). To extend th is idea to the present, more abstract, set up, we

assume that one may append to our balance law (6.1) an "entropy" inequa l i t y

_> S ° + R , (8.1)

where the entropy n, entropy f l u x ~ and entropy production R are determined by

via cons t i tu t i ve type re lat ions

n = n(~) , S ~ : S~(~) , R : R(U,X,t) (8.2)

A solut ion of (6.1) w i l l be cal led admissible i f i t sa t i s f ies (8.1). I t is standard

pract ice in continuum physics to require that a l l smooth solut ions be admissible

so we also impose th is condit ion here, assuming that

118

an [JB aSCZ + R (8.3) aU--~ = ~ T B UB,~

for every smooth solut ion U(X,t) of (6.4). This assumption induces the existence of

an n-vector valued integrat ing factor

: L(~) (8 .4 )

with the property

~n = L A aVA aU N aU N , (8.5)

aS s = L B ~PB aU N DU N , (8.6)

R : L A H A ( 8 . 7 )

In the example (6.3) of homogeneous hypere las t i c i t y ,

1 I~[ 2 - W(~) (8 .8 ) ~ : -- ~ PO

S ° = - viT~ (8.9) 1

R : - PoVlbi , (8.10)

L Im~ T I ~3 1 2 3, = - ,L3,v ,v ,v ) (8.11) ~'1' 2 . . . . .

so that the physical in terpretat ion of "entropy" is minus mechanical energy.

Our objective is to invest igate uniqueness and s t a b i l i t y of admissible solu-

t ions by using the methodology of D. Perna 131.

Since U + V (U) i s a diffeomorphism, we may v isual ize U,~,n,S m' L as functions

of ~ and H,R as functions of (~ ,~ , t ) . Then (8.5), (8.6) take the form

an - L M (8.12) aV M

aS s = L B ~P~ aV M aV M (8.13)

119

We now define

(8.14)

Assume that ~(X, t ) is a smooth ( i . e . L ipschi tz continuous) solut ion of (6.1) and

~(~ , t ) is any admissible solut ion of (6 .1) , of bounded var ia t ion . Using (6.1) , (8.1)

and (8.3) we deduce

> R- R- LA(HA - HA)-~A(VA- VA) +LA ( ~ - P~) (8.16) - m S A= D ~

By v i r tue of (6.1)

~A @LA VB @LA @P~ ~LA = 3V B - @V B @Vr I VM,~ + ~ T B HB

However, on account of (8.12) and (8.13)

(8.17)

aL A aL B : (8.18)

aV B ~V A ,

Thus (8.17) y ie lds

aL B aP~ aL B ~PB

aV A aV M aV M aV A

~A aL B ~PB @L B = @V M aV A VM,~ + @TA H B

Furthermore, from (8.7) ,

aPt L B aL B = ~ T A ,~ + ~TA HB

(8.19)

(8.20)

-R-LA(HA - HA) = HA([A - L A) + (HA - HA) ([A _ L A) (8.21)

Combining (8.16), (8.20), (8.21) and interchanging appropr iate ly the dummy summa-

t ion indices A and B we obtain

_ aP~ _ LA D ~ { - ~ _ _ ,~ - ,= ~ PA PA ~T B (VB VB)}

+ (H A - H A ) (L A - LA). (8.22)

+ {-L A _ L A aL A -~T B (VB-VB) }H A

120

The cruc ia l observation is that A and D m, as given by (8.14) and (8.15), as well as

the r ight-hand side of (8.22) are of quadratic order in ~ - V. Suppose that the

f low of D through the boundary of the body vanishes ( in the hype re las t i c i t , case

th is condi t ion w i l l be sa t i s f ied when, for example, the forces excerted upon the

boundary are equal or when the boundary moves with the same ve loc i t y in the two so-

l u t i ons ) . Assume, fu r the r , that A(V,V) is negative de f i n i t e . Then, in tegra t ing

(8.22) over the body and using Gronwall 's i nequa l i t y , one eas i ly arr ives at an est i

mate of the form

IBl ( ,t) - ~ K e kt IBIV(X,O) - ~(~,0)12 dX , (8.23)

where K and k are pos i t ive constants depending on the so lu t ion V(X, t ) . In par t icu-

l a r : when ~V(X'O)~ = V(X,O), XGB, the two solut ions coincide. We have thus shown

that , under the negative def in i teness assumption on A, whenever a smooth so lu t ion

of (6.1) ex is ts then there is no other admissible so lu t ion w i th in the broader class

of funct ions of bounded va r ia t i on , that sa t i s f i es the same i n i t i a l and boundary con

d i t i ons .

In view of (8.14) and (8.12), A w i l l be g loba l l y negative de f i n i t e i f and only

i f n is a uni formly concave funct ion of V. In the appl icat ions to continuum physics,

th is condi t ion is sometimes sa t i s f ied and sometimes not. For example, in the hyper-

e l a s t i c i t y case, n, as expressed by (8.8) , w i l l be concave i f and only i f the

s t ra in energy W is a convex funct ion of deformation gradient ~. I t can be shown,

however, tha t , due to the representation (5.4) , d ic tated, as we have seen, by the

p r inc ip le of frame ind i f fe rence, W can n ever be a g loba l l y convex funct ion of ~.

Even so, i t turns cut that W is convex on a large region in phase space, so that we

at least have local uniqueness and s t a b i l i t y of smooth solut ions with range in that

region of convexity.

In the most common case where no smooth so lu t ion of (6.1) ex is t the entropy

admiss ib i l i t y c r i t e r i on is no longer s u f f i c i e n t l y powerful to s ingle out the physi-

ca l l y relevant so lu t ion by ru l i ng out a l l extraneous ones. Stronger admiss ib i l i t y

c r i t e r i a that have been proposed include L iu 's shock adm iss ib i l i t y condit ion 141,

the v iscos i ty c r i t e r i on and the entropy rate c r i t e r i on I51. The entropy rate c r i t e -

r ion requires that admissible solut ions maximize at every point the expression

- S ~ R, i . e . , whenever there are many solut ions that sa t i s fy the entropy ine-

qua l i t y (8.1) then the admissible is the one that maximizes the rate of entropy pro

duct ion. The compat ib i l i t y of the various admiss ib i l i t y c r i t e r i a has been esta-

bl ished in special examples but the general problem is s t i l l open.

121

REFERENCES

111Truesdell, C.A. and Toupin, R.A., The Classical Field Theories. Handbuch der Physik (S. FlUgge, Ed.), vo l . l l l / 1 . Berlin: Springer-Verlag 1970.

121Truesdell, C.A. and Noll, W., The Non-Linear Field Theories of Hechanics. Hand- buch der Physik (S. Fl~gge, Ed. ) ,vo l . l l l /3 . Berlin: Springer-Verlag 1965.

131Diperna, R.j. Uniqueness of solutions to hyperbolic conservation laws (to appear).

141 Liu, T.P., The entropy condition and the admissibil ity of shocks. J. Math. Anal. Appl. 53, 78-88 (1976).

151 Dafermos, C.M., The entropy rate admissibil ity cri terion in thermoelasticity. Rend. Accad. Naz. Lincei, Set. VI I I , 5_7_7, 113-119 (1974).

MATHEMATICAL ASPECTS OF CLASSICAL NONLINEAR FIELD EQUATIONS

W. Strauss

Brown Univers i ty , Providence R.I .

Table of Contens

O. Introduct ion

I . The Nonlinear Schr~dinger Equation

I I . R e l a t i v i s t i c wave equations

I I I . Conservation laws

IV. Nonlinear Scattering Theory

Page

124

125

130

135

141

124

INTRODUCTION

Most of the c lassical nonl inear f i e l d equations are far too d i f f i c u l t for ma-

thematical analysis at th is time. However in recent years there have been some im-

portant advances for the simples equations. For th is reason I w i l l emphasize in

these lectures the model equations:

~U i W - Au + F(u) = O (NLS)

the nonl inear Schr~dinger equation, and

~2---~u - Au + m2u + F(u) = O ~t 2

the nonlinear Klein-Gordon equation. Here A = ~2/sx~ + . . .

x = (x I . . . . . x n) is in n-space R n, and F(O) F'(O) = O.

(NLKG)

~2/~x2 is the Laplacian, n

I t is usual ly easy to solve such equations for a short time (or " l oca l l y in

time" as we say) but we are interested in global questions, l~hat general properties

does the complete time evolut ion of the system have and in pa r t i cu la r what is i t s

asymptotic behavior?. Spec i f i ca l l y nonl inear phenomena only become evident in the

long-time behavior. These include under various circunstances: ( i ) the development

of shock waves (not for NLS or NLKG but for the equations discussed by Prof. Dafer-

mos in his lec tures) , ( i i ) blow up of solut ions in a f i n i t e time, ( i i i ) the exis-

tence of so l i tons , a kind of nonl inear general izat ion of bound states, and ( iv ) re-

l a t i v i s t i c scat ter ing phenomena. I t must be emphasized that solut ions of nonl inear

equations cannot be combined l i nea r l y to obtain new solut ions and in par t i cu la r

they cannot be constructed by Fourier analysis.

The plan of these lecture notes is as fo l lows. In chapter I we discuss NLS,

i t s appl icat ion to the theory of lasers, and i t s bound states. In chapter I I we

discuss NLKG and other r e l a t i v i s t i c wave equations and the general question of the

existence of so lut ions. In chapter I I I we discuss the conservation laws which fo-

l low from the invariance propert ies of the equations and apply them in par t i cu la r

to the Yang-Mills equations. In chapter IV we discuss scat ter ing theory mainly in

the context of NLKG. The references l i s ted are not meant to be complete or even re-

presentative of the topics covered.

125

I . THE NONLINEAR SCHRODINGER EQUATION

1.1 Trapping and focusing of laser beams

Consider the electromagnet ic wave equation

I 22 (EE) - AE = 0 ( I ) c 2 ~t 2

Assume a l i n e a r l y po lar ized wave (~ pa ra l l e l to a f i xed u n i t vector ~) which is mo-

nochromatic w i th frequency m and which propagates along the z -ax is . Thus

: u ( x , y , z ) exp( i (kz - ~ t ) )

and (1) reduces to

~U 2ik ~ - Au + (k 2 - ~ 2 / c 2 ) u = 0 (2)

The high i n t e n s i t y of a laser beam can produce s i g n i f i c a n t local changes in the den

s i t y of the medium and hence in the d i e l e c t r i c constant ~.

Chiao, Garmire and Townes 111 assume the simple non l inear dependence + 2

E = t o + s2 IEI " They show how the r esu l t i ng nonl inear term may give r i se to an

electromagnet ic beam which produces i t s own waveguide and propagates w i thou t sprea-

ding. This phenomenon is ca l led " s e l f - t r a p p i n g " . I t corresponds to a so lu t ion of

(2) which is independent of z:

_ m2 2u = - ~ 2 ~ u - ~2--~u + (k 2 t o ~ ) u - E 2 - Z l ul 0

~x 2 ~y2 c c (3)

Kel ley and Talanov 121 show how a non l inear dependence of ~ can produce a

bu i ld -up in the i n t e n s i t y of par t of the beam as a func t ion of z. This phenomenon is

ca l led " se l f - f ocus i ng " . I t corresponds to a so lu t ion of (2) in which the i n t e n s i t y

lul 2 blows up at a cer ta in value of z. I f the "parax ia l " approximation fUzz I << IkUzl

is used and we choose k = ~7-~o/C, equation (2) reduces to

~u ~2u E 2 k lulZu = 0 (4) 2ik T ~ - sx 2 - sy2 - ~o

Under appropr iate assumptions about the nature of the absorbing medium, the

beam can be spread out , d i s to r ted or bent instead focused. These phenomena are

ca l led "thermal blooming". I f we take in to account the e f fec ts of a steady t rans-

126

verse wind along the y - a x i s , then ~ - o

upstream. In the s implest case we have

depends on the beam i n t e n s i t y lul 2 ups

rY - ~o : const I l u l2d t (5)

/

A s i m i l a r equat ion describes the long- t ime behavior of one-dimensional hydro-

magnetic waves in a plasma:

~u ~2u 12 i ~ - - c - - - (a lu + b)u : 0 (6) ~x 2

where a,b,c are constants. Such an equat ion also occurs in the Ginzburg-Landau theo

ry of superconduc t i v i t y .

1.2 Existence and blow-up

We consider the problem

~ - Au +~lul p - I u = 0 i

u(x,O) = ~(x) , xGIR n

(7)

where @ is a nice function (@G S i f not stated otherwise), p > 1 and ~ is real.

The two standard conservation laws are obtained by multiplying (7) by u and taking

imaginary parts, and by multiplying by ut and taking real parts. They are

I lul 2 dx = constant and

r ~ p+l E = I {½ Ivul 2 + ~ lul }dx = constant .

They would lead us to be l ieve tha t so lu t ions ought to e x i s t g l oba l l y and be stable

i f ~ > O, but tha t i f ~ < 0 i n s t a b i l i t y may be possib le.

Theorem 1.1 There ex is ts a unique so lu t i on in some time i n te r va l {xGIR n, I t l < t l } .

I t is as smooth as the s i n g u l a r i t y at u=O of the func t ion lul p - I u a l lows; i f p

is an odd in teger i t is C =.

This local existence r e s u l t fo l lows eas i l y by standard methods (See Sect ion

I1.4).

127

Theorem 1.2 Let x < 0 and p ~ 1 + 4/n. Let , sa t i s f y the cond i t ion

f ½ 12 x ip+Z E: ] { Iv, dX+p-TTl* }dx<0

Then no smooth so lu t ion of (7) can ex i s t fo r a l l t .

In case p =3, n =2, the equation reduces to (4) and Theorem 2 provides a pre-

c ise i n i t i a l cond i t ion fo r the se l f - focus ing o f an electromagnet ic beam. The oppo-

s i t e s i t u a t i o n is described in the next theorem.

Theorem 1.3 I f x < 0 assume p < 1 + 4/n. I f x > 0 and n > 2 assume p < 1 + 4 / ( n - 2 ) .

I f , 6 H l ( R n ) , there ex is ts a unique so lu t ion of (7) fo r a l l t which is a bounded

continuous funct ion of t wi th values in H l ( R n ) o The r e g u l a r i t y statement of Theo-

rem 1 is va l i d fo r a l l t .

In p a r t i c u l a r , equation (6) has global smooth so lu t ions .

In view of Theorems 2 and 3, we ca l l ~ > 0 and X < O, p < i + 4/n the stable

cases and we ca l l x < O, p ~ i + 4/n the unstable case.

Theorem 1.4 Let ~ > 0 and p < =. I f ,GH I (~n) , there ex is ts a so lu t ion which is a

bounded, weakly continuous funct ion of t wi th values in 111(~n).

This theorem is proved below. We do not know whether th is weak so lu t ion is

unique or smooth.

Theorem 1.5 For any x, there ex is ts a unique smooth so lu t ion o f

i ~-~u ~x 2~2y _ ~y2~2Y + ~u iY l u ( x , y ' , t ) 12dy' = 0

wi th u(x,y,O) = , ( x , y ) . (See 131).

Proof o f Theorem 1.2 Write F(u) = x lu ] p-1 u and G(u) = ~ l u l P + l / ( p + l ) . Mu l t i p l y

+ nu where r = Ixl and take real par ts : equation (7) by 2ru r

I rUr ~ il fl dt Im dx = - 2 vu l 2 dx + n 2G(u) - uF(u) dx

= - 4E + I (2n + 4) G(u) - nuF(u) dx

- 4E since p ~ i + 4/n

128

Now mu l t i p l y (7) by r2u and take imaginary parts:

d Ir21ul2dx = _ 4 Im irUrG dx

Hence

02 i • r21ul2dx < 16E < 0 dt 2

This is nonsense because the l as t in tegra l is pos i t i ve .

Proof of Theorem 1.3 ~e merely sketch the ingredients . ( i ) I f p < 1 + 4 / ( n - 2 ) , So-

bolev 's theorem states that H I L p+I. ( i i ) The evolut ion operator Uo ( t ) : , ÷ u( t )

fo r the f ree ( l i nea r ) Schr~dinger equation s a t i s f i e s the estimate

t-~llml llUo(t)~ll q lq,

fo r 2 ! q ! ~, 1/q + 1/q' = 1, 6 = n/2 - n/q, as can be obtained by in te rpo la t ion

from the cases q =2 and q = ~. I f q = p + l , then 0 < 6 < i so that t -6 is in tegrable

at t =0. ( i i i ) We use the conservation laws. I f x>O, both terms in E are non-nega

t i v e and hence bounded for a l l t . I f I <0, Sobolev's i nequa l i t y states that

I l u lP+ ldx <_ c { I Iut2dx}°~{, I lVul2dx}l~

where 2~ = ( 2 - n ) p + 2 + n and ~ = n ( p - 1 ) / 4 . I f p < l + 4 / n , we have ~<0 and #<1.

Hence the conservation laws provide a bound in H I and in L p+I. ( i v ) To prove the

uniqueness, l e t u and v be two so lu t ions. By ( i i )

r t f lu( t ) - v ( t ) IIp+Z~ c J o ( t - s ) -~ II lul p - l u - Iv l p - l v l l ( p + l ) / p ( s ) ds

~t c l o ( t - s ) -~ I I u - Vllp+l(S) ds

since u and v are bounded in L p+I, by ( i i i ) . This impl ies u =v because ~<1. (v) The

existence and regu la r i t y are proved by combining ( i ) - ( i i i ) with Theorem I .

Proof of Theorem 1.4 Let g(s) = x s P + l / ( p + l ) fo r s ~ O. We approximate g(s) by a 2 for s large and sequence of smooth funct ions gN(s) such that gN(s) = const s

0 ~ gN(s) ~ g(s) , gN(s) ÷ g(s) fo r a l l s. Let F(u) = ~ lu lP- lu and FN(U) = I

= gN( lU l )U/ lU l . Let uN(x, t ) be the unique so lu t ion of the problem

iuNt -Au N + FN(u N) = 0 , uN(x,O) : ~(x) (8)

129

I t ex is ts because FN(U) is l i n e a r fo r large u so that an easy va r i a t i on of Theo-

rem 3 is app l i cab le . Then we have the obvious a p r i o r i bounds

il urtl2dx : i1¢12 dx and

I {½ IvuNl 2 + g,(lu, l)}dx [{½ Iv l 2 + g(1 l)}dx )

By compactness, there is a subsequence ( s t i l l ca l led UN) which converges to some u

weakly in H 1 and almost everywhere, Hence gN(lUNl) ÷ g ( l u l ) a.e. and FN(U N) ÷ FN(U)

l o c a l l y in L I , by Theorem 1.1 of 141. We may pass to the l i m i t in each term in (8).

The other proper t ies o f u fo l l ow eas i l y as in 141.

1.3 Bound States

I f we subs t i tu te a standing wave u = ¢(x) exp ( - i~ t ) in to NLS, we get the e l l i R

t i c equation

me - A¢ + F ( ¢ ) = 0 (9 )

provided that arg F(u) = arg u. This is the same equation as (3) fo r a p a r t i c u l a r

F. Into NLS we can also subs t i tu te the more general expression

u = ¢ ( x - c t ) e x p [ i g ( x - b t ) ]

where ¢ and g are real sca lar funct ions and c and b are constant vectors. This is a

so lu t ion of NLS provided ¢ s a t i s f i e s (9) and g(x) = -c .x /2 + go' go = constant and

4~ = c . ( c - 2 b ) > O.

We def ine a general ized bound state as a so lu t ion of NLS which has f i n i t e ener

gy ( is square- in tegrab le) fo r each t but whose amplitude max l u ( x , t ) l does not go to X

zero as I t l ÷ ~. Then we are that every so lu t ion of (9) leads to a whole fami ly

of general ized bound states of NLS depending on several parameters. Here are some

examples of equations o f the form (9). We take n =3. By sca l ing ( x + c x and ¢÷c¢)

we can adjust the values of various constants; f o r instance we can always assume

= I in (9). We only consider so lu t ions which vanish at i n f i n i t y and which are not

i d e n t i c a l l y zero.

130

The e q u a t i o n -&@ + @ - I~I q-1 ~ = 0 possesses an i n f i n i t e sequence o f d i s -

t i n c t s o l u t i o n s i f 1 < q < 5, bu t no s o l u t i o n s ( o f f i n i t e energy ) i f q Z 5. These

s o l u t i o n decay e x p o n e n t i a l l y as Ix l . ÷ ~. The e q u a t i o n -A@ - ~ + ~sin@ = 0 possesses

a t l e a s t one ( n o n - t r i v i a l ) s o l u t i o n i f x > I . The e q u a t i o n

. A¢ + @ + I@IP-I@ - ~ l @ l q - l ¢ = 0

has at least one solution i f i < q < max (p,5) and ~ > ~, where ~, = ~,(p,q) is e~

p l i c i t l y computable. The equation -A@ - I@I q-1 = 0 possesses a solution only i f

q = 5 in which case @(x) = c(c 4 + Ix12) -I /2 is one. The equation

-a@ + I@IP-I@ - l@lq-1@=O hasasolution i f 5 < q < p and i f p < q < 5. For another

exp l ic i t example, see the lectures of Prof. Bialynicki-Birula.

Many of these states are unstable, but some of them appear to be stable al-

though this has not yet been proved. (There are many definitions of s tab i l i t y and

so one has to be very careful). As references to the assertions above, see 15,6,71

I I . RELATIVISTIC WAVE EQUATIONS

I I .1 The nonlinear Klein-Gordon equation

There is a well-developed theory which is quite analogous to NLS, and so we

just state the facts. The most basic invariant is the energy:

l 2 +½1vul2+ 21_m 2 u 2 + G(u) dx = const (1) E=I ½ut

where G(u) = We cons ider the Cauchy problem: NLKG toge ther with the i n i -

t i a l (or Cauchy) condi t ions

u(x,O) : f(x) , ut(x,O) : g(x)

The functions f(x) and g(x) are given functions which are arbitrary with in a cer-

tain class.

Theorem 2.1 There exists a unique solution in some time interval I t l < t 1, where

t I may depend on the size (of some norm) of f and g.

Theorem 2.2 I f F(u) = O(lul 2) as u ÷ O, the solution can be extended to al l times

- ~ < t < = provided the i n i t i a l data is suff ic ient ly small (in some norm).

This is proved by observing that, for small solutions, the term IG(u)dx in the )

131

E can be bounded by the term lu2dx. Hence each term in (1) is bounded for energy 2

a l l t ime.

Theorem 2.3 For a r b i t r a r y i n i t i a l data of f i n i t e energy, there ex is ts at leas t

one weak so lu t i on fo r a l l time provided uF(u) ~ - c ( l + u 2) and G(u) ~ -c

fo r some c > 0 and a l l u. A major open problem is the uniqueness under these condi-

t i ons .

Theorem 2.4 The so lu t ion in Theorem 2.3 is unique i f IF(u)Igrow s t r i c t l y slower

than lul 5 as lul ÷ ~ . In th is case the so lu t ion is as smooth as F, f and g a l lows.

Theorem 2.5 I f uF(u) ~ (2+s)G(u) fo r a l l u and some s < O. I f there ex i s t i n i t i a l

data with energy E < 0 and _ [uutdx > 0 at t = O, then the so lu t ion (of Theorem 2.1)

wi th those data blows up in a f i n i t e t ime.

Example 1. F(u) = sin u. By Theorem 2.4 there ex is ts a unique global C ~ so lu t ions

fo r a r b i t r a r y C ~ i n i t i a l data.

Example 2. F(u) = - ] u l P - l u wi th p > 1. By Theorem 2.5, many so lu t ions blow up in a

f i n i t e t ime.

Example 3. F(u) = + lu lP - lu . I f 1 < p < 5, there is a unique global so lu t i on fo r

a r b i t r a r y i n i t i a l data (by Theorem 2.4) . I f p ~ 5, each pa i r of i n i t i a l data gives

r i se to a global weak so lu t i on (by Theorem 2.3) but i t is unknown whether th is so-

l u t i o n is unique or not. Numerical computations ind ica te that i t is probably unique.

They also show tha t , as p increases wi th the same i n i t i a l data, the amplitude of

the so lu t ion decreases and the number of o s c i l l a t i o n s increases. Thus the e f f e c t o f

a high power is to throw more of the energy in to k i ne t i c energy.

Some references fo r the above discussion are 18, 9, 101 .

11.2 The Maxwell-Dirac Equations of QED

They are

(MD)

I ( i ~ - M ) ~ = g ~ ~

F ~ = g ~ yP~ 3X v

~A ~ ~A v F ~1~ =

~X ~X "~ p

charge f ] ~ + ~ d x i s conserved. The energy is conserved. In the Coulomb gauge The

3 ~A k Z ~x k - 0 ,

k=l

132

i t takes the form

e(A) = I [½ ~ { (3A_~)2+ ivAki 2 } + ½ ivAOl2 ]dx k=l

e(A) + M~}, - Im ~ dx = cons tan t . k=l

fWarning: In some o ther gauges, e ( A ) w i l l have some negat ive terms] . The fo l lowing I . - J

es t imate can be deduced II11:

( 12dx)I/2 e(A) 5 const ( I + Jlv (2)

11.3 The Yan9-Mil ls Equations

8

We take the metr ic -+++ in space-time and the gauge group G = SU(2). We wr i te

= ~/~x (u = 0,1 ,2 ,3) fo r the ordinary calculus der i va t i ves and

Dk = ~k + g AkX (k = 1,2,3)

Do : ~o - g AoX (k = O)

fo r the covar iant de r i va t i ves . The gauge potent ia ls A are funct ions on space-time

wi th values in the Lie algebra of G. Of can be regarded as ordinary 3-space and the

A as a vector f i e l d on space time (~ = 0 ,1 ,2 ,3 ) . The f i e l d strengths may be de f i -

ned by the operator equations

g F x = D D - D D (~,~ = 1,2,3)

g F o x = DoD - D D q o

Let E k = Fko (k = 1 ,2 ,3) , H 1 = F32, H 2 = F13, H 3 = F21 be the e l e c t r i c and magnetic f i e l d strengths. Let E be the 3 x 3 matr ix whose columns are EI,E 2 and E 3. Let H be

the 3 x 3 matr ix with the columns H I , H 2 and H 3. The Yang-~Hlls equations are the

equations of motion for the Lagrangian densi ty

: ½ IEI 2 - ½ qHk 2

The equations are (YM):

DoE I = D2H 3 D3H 2

DoE 2 = D3H 1 DIH 3

133

DoE 3 = DIH 2 - D2H 1

DIE I + D2E 3 + D3E 3 = 0

The i r resemblance to Maxwel l 's equations is more than co inc iden ta l . In f ac t , they

are simply Maxwel l 's equations su i t ab l y modif ied to be i nva r i an t under local gauge

t ransformat ions. There are also four " cons t ra in t " equations which fo l l ow from the

d e f i n i t i o n s of D , E k and H k. They are

DoH I = D3E 2 - D2E 3

DoH 2 = DID 3 - D3E 1

DoH 3 = D2E 1 - D I E 2

DIH 1 + D2H 2 + D3H 3 = 0

From these e igh t equations i t fo l lows that the energy is

2 i ( IEI2 + IHl2)d3x = constant.

See chapter I I I .

We shal l not discuss at a l l the "Eucl idean Yang-Hi l ls equations" wi th the me-

t r i c ++++, fo r which there has been so much recent progress. Un fo r tuna te ly , on the

c lass ica l level the Euclidean and Minkowski versions appear to be unrelated.

11.4 General d iscussion of the existence question

The r e l a t i v i s t i c equations j us t discussed can a l l be put in to the form

Lu = Nu , u(O) = ~(x)

where u(O) denotes the i n i t i a l data at time t=O, u is a vector func t ion , L is a l i -

near hyperbo l ic pa r t i a l d i f f e r e n t i a l operator , and N is a non l inear operator in u

but not in the de r i va t i ve of u.

One theorem is r e l a t i v e l y easy: there is a unique so lu t i on in some time i n t e r -

val I t l < t I where t I depends on I I~ i l . (See Theorem 2.1) . There is a great deal o f

f l e x i b i l i t y in choosing the norm II II. For MD i t has been car r ied out in I121 and

fo r YM in 113 i . For the E ins te in equations see 1141.

134

The global existence problem is: do arb i t ra ry i n i t i a l data # launch solutions

which ex is t for a l l time?. A possibly easier problem is to res t r i c t the size of #.

Obviously this problem must have an a f f i rmat ive solut ion (at least for a rest r ic ted

class of #) before any scatter ing problems can be considered. There are two methods

for attaching the problem.

Method 1. Solve the equation from time 0 to time t 1. (For this step we need a

strong enough norm If#If). Use the solut ion at t lme t I as Cauchy data and solve up

to time t 2. Then solve up to time t 3 and so on. The method succeeds i f the se-

quence of times t l , t 2 . . . . tends to i n f i n i t y . Now t I depends on the size of l l#II;

t 2 - t I depends on the size of I lU ( t l ) I I ; and so on. Therefore the problem would be

solved i f we knew that

IIu(t)II constant

for whatever times t the solut ion may ex is t . (Actual ly we only need to know that

l lu ( t ) l l is bounded for bounded t . ) Mathematicians cal l th is an a p r io r i estimate.

Method 2. Approximate the equations. Find global solut ions u of the approxi- c

mate problems. Thus L~u~ = N u ,u (0) = # . Then pass to the l i m i t and t ry to

prove that u converge to a solut ion u of the or ig ina l problem. The key step is

again an a p r i o r i bound, say llluElll -< constant independent of ~. Then we can take

weak l im i ts u ÷ u. I t is usually easy to show that L u ÷ v, that N u ÷ ~, and

that v = Lu. Thus Lu = w. The d i f f i c u l t y in this method usually is to prove that

w = Nu. (Easy examples show that i t can happen that w ~ Nu). For this we again need

to use a strong enough norm III III- As examples of approximation procedures we men-

t ion: (a) truncating the nonlinear terms, (b) adding viscous effects (see Prof. Da

fermos's lectures) , or (c) approximating by f in i te-d imensional problems as in nume-

r ica l analysis.

They key point in both methods is the a p r i o r i estimate, which depends very

precisely on the structure of the equations. The signs of the nonlinear terms are

c ruc ia l , as we already saw in the cases of NLS and NLKG.

For the MD system, we know that I I~I2dx is bounded for a l l time. This es t i - Y

mate is not strong enough. The estimate (2) shows that we get a good bound on A i f

could get bounds on - {Iv~12dx. No one has found a way to accomplish th is . But at w e

least in the one-space-dimensional analogue of MD i t has been carried out (see 1151).

For YM the existence question is completely open ( r e l a t i v i s t i c case).

Very recent ly some s ign i f i can t progress has been made on the global existence

problem for small data ~ by S. Klainerman 1161. He considers essent ia l ly a rb i t ra -

135

ry (~) quadratic nonl inear perturbations of the ordinary wave equation u t t = Au,

xGIR n . I f n _ > 5 he proves a resu l t l i ke Theorem 2.2 above. The dimension enters

because there is more spreading in higher dimensions. He uses the very sophist ica-

ted Nash-Moser approximation technique.

See Prof. Dafermos' lectures for fu r ther discussion of the existence question

and an existence proof for some h igh ly nonl inear systems.

I I I . CONSERVATION LAWS

111.1 The Euclidean equation

Au = F(u(x)) , xGIR N (1)

We assume F is a real funct ion such that F(O) = 0 and u(x) is a smooth real func-

t ion going to zero as Ixl + ~.

Equation (1) can be wr i t ten va r i a t i ona l l y as aE[~ = O, where

I Z lvul2 E[u] = { ~ + G(u)}dx

energy and G(u) = I~ F(v)dv. This can be expressed formal ly as fo l lows. Let is the

T~ be a smooth family of transformations such that T o I . Let H = dT /d~ at ~ = O.

For any funct ion u = u(x) ,

d-~ ~= o E[T u] = (E ' (u) , Mu)= ( -vu+F(u) ,Mu).

Here M stands for "mu l t i p l i e r " . I f u is a solut ion of (1), th is expression va-

nishes. This i l l u s t r a t e s the general p r inc ip le of Noether (1918) that i f a one-pa-

rameter family of transformations leaves a var ia t iona l problem invar ian t , the solu-

t ion sa t i s f i es a conservation law. In our case i t means that the product

(-vu + F(u))(Mu) is a divergence.

I t is well-known that the Laplace operator is invar ian t under the conformal

group ~ , the group of transformations on IR N which preserve angles. I f N ~ 3, th is

group consists of four types of transformations: t rans la t ions , ro ta t ions, d i l a t i on

and inversions. The to ta l dimension of ~ is therefore N (N- I ) / 2+2N+ I (= 15 i f N=4~

On the other hand, equation ( I ) is invar ian t only under the Gali lean group but not

under the whole conformal group, with the exception of one par t i cu la r F. We propose

to exp lo i t th is fac t , looking separately at the various generators o f ~ .

136

The t r a s l a t i o n T : u(x) ÷ u ( x + ~ a ) , where a is a constant vector , has M=a.v

as i t s i n f i n i t es ima l generator. Wri t ing ( - vu+F(u ) ) (Hu) as a divergence, we get the

conservat ion law

0 = v . { - (a . vu )vu + a ( Ivu I2 /2 + G(u))}

We get N independent laws by choosing a as the un i t vector in the coordinate d i rec -

t ion Xk:

0 = {-u~ + T1 ivul2 + G(U)}k+ j~k{_UjUk} j (2)

where subscr ipts denote par t ia l de r i va t i ves .

The ro ta t ions give the N(N-1)/2 mu l t i p l i e r s XkU j - x ju k fo r j ~ k and the con-

servat ion laws

0 : v.{(-XkU j + XjUk)VU} + {Xk( IVul2/2 + G ( u ) ) } j - { x j ( I v u [ 2 / 2 + G(u))} k. (3)

The d i l a t i o n u ÷ ul leaves the D i r i c h l e t in tegra l i nva r i an t , where u~(x) = v m+l = Imu(xx). To f ind the cor rec t value of m, we ca lcu la te u1(x) = I ( vu ) ( lu ) and

1 12 l E[ux] = I { ~ x 2 m + 2 l (vu) (xx ) +G(xmu(xx)}dx = ] { } x2m+2-N lvu (y ) 12+x-NG(xmu(y))}dy

where y = xx, dy = xNdx. The f i r s t term is i nva r ian t i f 2m+2-N = 0 or m = (N-2)/2.

For th is choice of m,

o: [uj ,01u)•

The m u l t i p l i e r is

Mu = ad-~Imu(xx) x=l = x.vu + mu

The conservat ion law is

o = ~ uF(u) .G(u)+v {(x vo)vu + ½ xLvui2 + ~ uvu+ xG(u)~

Equation (7) provides some n o n - t r i v i a l in format ion about possible so lu t ions

of (1). We have

(5)

137

( i f N }~ 2). Therefore

E1ul = Ilvul2dx->O

The f o l l ow ing theorem fo l lows eas i l y .

Theorem 3.1 I f u is a so lu t i on of (1) , smooth and zero at i n f i n i t y , then the ener-

gy is pos i t i ve (except i f u z 0 ) . There can be no so lu t i on of (1) i f any one of the

f o l l ow ing four funct ions is pos i t i ve ( fo r s ~ 0):

sF(s) , G(s), H(s), -H(s)

where we assume N f 1, H(s) = (N-2)sF(s) - 2NG(s).

The one-dimensional case (N = 1) is t r u l y except ional since i t permits so lu-

t ions even i f G 3 O° The theorem is due in par t to Derr ick 171 and in par t to

Strauss 151.

We have seen above tha t the non l inear equat ion is not i n v a r i a n t under the trans

formation u ÷ ux. However, i t is i n va r i an t in the special case

-NG(u) + - ~ u F ( u ) = O, G' : F.

That i s , G(u) = const u 2N/(N-2). In t h i s case, our va r i a t i ona l problem is equiva-

l en t to f i n d i n g the best Sobolev constant I[@II2N/(N_2 ) ~ const llv@l12. See Strauss

151. The invers ion V: x ÷ x / x . x is the fo r th kind of conformal t ransformat ion. I t

leaves the u n i t sphere ix l 2 = 1 i n v a r i a n t and V 2 = I . I f we l e t

v(x) = [ x l 2 - N u ( x l x [ - 2 ) , a ca l cu la t i on shows tha t I l v v ( x ) 12dx = I l v u ( y ) I2dy. An

N-parameter fami ly of invers ions is given by y = Va(X) where

y / l y l 2 = x/IxE 2 + a (a G IR N)

That i s , V(y) = TaV(X ) where T a is t r a n s l a t i o n . So we may wr i t e V a = VTaV or

x+alxl 2 y = Va(X ) = _

Z + 2 a . x + la l21xl 2

These invers ions given us N ra ther complicated conservat ion laws. The m u l t i p l i e r s

are e s s e n t i a l l y

= Ixl2a.vu - 2(a.x)(x.vu) ~-T u(V~a(X)) ~=o

Prec ise ly , they are

138

Mu = [ Ix l2a - 2 x ( x - a ) ] - v u - (N-2)(x-a)u (6)

We leave as an exerc ise the w r i t i ng of the invers ional conservat ion laws.

111.2 NLKG.

We Lake n space dimensions. We can t rans fe r each of the Euclidean i d e n t i t i e s

by making the fo l lowing changes:

N = n + l ,

x ÷ (Xl,X 2 . . . . . X n , i t ) , x N = Xn+ I = i t

F(u) ÷ m2u + F(u)

Thus there are (N 2 + 3N + 2)/2 i d e n t i t i e s which immediately fo l low from t h e i r Eucli

dean counterparts. Here they are, a f t e r i n teg ra t i on over space coordinates only.

From the m u l t i p l i e r u t = ~t u, we get the energy

le (u )dx : I (½ u~ + ~ Ivu[2 + ½m2u2 + G(u)dx = constant.

From the m u l t i p l i e r u k = ~k u, we get the momenta

l utu k dx = constant.

We get the angularmomenta from the mu l t i p l i e r s XkU t + tu k and XkU j - XjUk:

I (xke(u) + = const. tukut )dx

and

I(XkU j - = co~st. XjUk)Utdx

The next two i d e n t i t i e s are due to Morawetz 1121: From the m u l t i p l i e r

Mu = tu t i d e n t i t y

where

+ ru r + ~ u, where r = Ixl is the spat ia l radius, we get the d i l a t i ona l

0 = d ~ i ( t e ( u ) + rUrUt + -~uu t )dx+21 -1H(u )dx

H(u) = ( n - 1 ) u F ( u ) - 2(n+l)G(u) - 2m2u 2

139

Final ly we get the inversional or conformal ident i t ies . From the mul t ip l ier

(k = N = n+Z in (16)) Mu = ( t 2 + r2)ut + 2rtu r + (n- 1)tu,

we get the ident i ty

0 = d -~ I [ ( t 2 + r 2 ) e ( u ) + 2 r t u r u t + ( n - l ) t u u t - n~__1_1 u2]dx + t IH(u)d x (7)

From the mul t ip l ie r (see (6))

Mu = txku t+~l ( t 2+2x~_r2)u k+x k Z X uj+n~---ll Xk u, j~k j

we get the ident i ty

0 : d-~I~Xk e(u)+½ (t2 +2x~-r2)uku t + x k ~ X-U-Ut + - ~ XkUt~dx+½ (])xkH(u)dx j~k J J

Another ident i ty due to Morawetz ]19] is obtained using the spatial radial der ivat ive as the mul t ip l ier . Thus

Mu = ~-~+ u, r = lxl .

A direct calculation shows

o : fu Ur+ u, x+

n # l I ( u F ( u ) . 2 G ( u ) ) ~ (8)

for n ~ 3, with the extra term 2~u2(O,t) in case n = 3.

111.3 The Yang-Mills equations

They too are conformally invariant (as are Maxwell's equations), so again there are 15 conservation laws. One fo them is the energy already wr i t ten in chap- ter I I . The momenta are ~ jpkdx = constant, where

1 H 2 E 3 E2.H 3 p = ° -

and p2, p3 defined s imi lar ly . Of the other conservation laws the most useful one is the inversional law analogous to (7). For YM i t takes the form

140

j [ i ( t 2 + r 2) ~ (IEI 2 + IHI 2) + 2t ~ xkP dx = constant k

I f we complete the square in this in tegra l , we see that i t is at least

(9)

( t - r ) 2 ( I E l 2 + IH12)/2

which is posi t ive[ In par t i cu la r , i f we r e s t r i c t the integral to any cone of smaller

aperture than a l i gh t cone, we obtain the fol lowing resu l t 1201.

Theorem 3.2 Let R > 0 and 0 < s ~ 1. As t ÷ = we have

(IEI 2 + -LH[ 2) dx = O(t -2) IxI<R+(I-E)t

This theorem means that a l l the energy moves along the l i gh t cone; that i s , at

uni t speed. I t follows that YM cannot possess a solut ion of the form E ( x - c t ) ,

H ( x - b t ) with Ibl < 1, Icl < 1 which vanishes at i n f i n i t y . Thus we can say that YM

possesses no generalized bound state. I t can also be shown that 12 of the 18 compo-

nents of a f i n i t e energy solut ion of YM are square integrable on any l i gh t cone.

For the de ta i l s , see 1201. The same resu l t is true for a Yang-Mills f i e l d coupled

to a Klein-Gordon f i e l d as in a Higgs model.

The Maxwell-Dirac equations in the mass-zero case are also conformally inva-

r ian t . The resul t ing conservation laws have been calculated in 1111.

111.4 NLS

We assume F(u) = g ' ( l u l ) u / l u l where g(O) = g'(O) = 0 and we l e t G(u) = g ( l u l ) .

I t has the Lagrangian

[

which is invar iant under the transformation u ( x , t ) ÷ xn/2u(xx,X2t) i f G = O. This

leads to the mu l t i p l i e r ru r + 2tG t + (n/2)u and the d i la t iona l iden t i t y :

d + [2 2tG(u~dx + [nF u)u - 2(n+2)G(u)Idx = 0 ~ t - I [ ½ Im(rUrU) t l vu + ½1 ( (I0)

Ginibre and Velo 1211 discovered the fol lowing analog of the inversional iden-

t i t y , which they cal l the pseudo-conformal conservation law. For the free

evolution operator,

XUo(-t)f = Uo(- t ) (x + 2 i t v ) f .

141

The i d e n t i t y is

d |2 4t2G(u)]dx + 2t i -2(n+2 _ = 0 (11)

Let F(u) = x l u | P - l u . I f p = 1 + 4/n, both (2) and (3) are exact conservat ion

> O, (11) impl ies tha t - ]G(u)dx = O( t -2 ) . laws. I f

IV. NONLINEAR SCATTERING THEORY

IV . I In the absence of bound s ta tes , one normal ly expects the asymptotic behavior

as t ÷ ± ~ to be f ree. Thus i f u ( t ) = u ( x , t ) is the i n t e r a c t i n g f i e l d , we look fo r

free f i e l d s u+( t ) and u_( t ) such tha t

f lu ( t ) - u± ( t ) l l + 0 as t ÷ ± ~ ( I )

The sca t te r ing operatQr S is def ined by

S: u_(O) ÷ u+(O)

For i l l u s t r a t i o n l e t us take the NLS equation

~u i ~ - Au + F(u) = 0 ( x g ~ n )

where arg F(u) = arg u. Let H o = + ~ be the free Hamil tonian. Then we def ine ( f o r -

mal ly fo r the time being)

i t i ( t_s)Ho u_( t ) = u ( t ) + _ ~ F(u(s))ds (2)

Formally u s a t i s f i e s the asymptotic property (1). By d i f f e r e n t i a t i o n of (2) we have

~u t f ~u + iH o J | + F (u ( t ) ) ~t ~ t _=" • •

au + iHo( u _u) + F(u) = iH u ~t - o -

which is the free SchrSdinger equat ion. We consider i t to act ion the H i l b e r t space ~f_= k2(~n) .

As a second i l l u s t r a t i o n , take NLKG. I t can be w r i t t en in Hamil tonian form as

fo l l ows . Let

142

<ul I ° :I I°°> = , i H o = A - m 2 ' ~ = u t 0

Then NLKG is equivalent to the "vector" equation

÷ ~ - iH ~ - F ( u ) ~t o

÷ The free Klein-Gordon equation is equivalent to ~ / ~ t = iHov. I t acts on the ener-

gy H i lbe r t space~/C. The norm is

II 112 : + Ivul 2 + m 2 lul2)dx

Thus the f i r s t component u of ~ belongs to L 2 and its spatial derivatives as well;

the second component u t of ~ also belongs to L 2. The solution of the free equation

exp(itHo) is a unitary operator on~[. The generator iH is a self-adjoint opera-

tor (but not taking a l l o f ~ i n t o ~F[). The in terac t ion ~ is a nonl inear operator

which may take a l l o f~ ) { i n t o~ [unde r special circunstances.

With th is notat ion equation (2) is the d e f i n i t i o n of ~ i f arrows (÷) are put

on ~ and ~. This is a 2 component in tegral equation. I f we wr i te i t s f i r s t compo-

nent, i t takes the form

I t u_(t) = u( t ) - I_ Dret ( t -s ) F(u(s))ds

Here Dre t is the retarded Green's funct ion for the KG equation. This is known as

the Yang-Feldman equation.

IV.2 Mathematical resu l ts .

The discussion which fol lows focuses on NLKG. The in tegra l in (2) converges

absolutely in the energy norm (norm o f ~ - ) provided

I° (IIF(u(x,t))12dx)i/2dt <

This leads na tu ra l l y to the question of the decay as I t l ÷ ~ of the solut ions of

NLKG. Now i t is well known that free solut ions (with s u f f i c i e n t l y nice i n i t i a l data)

decay to zero l i ke I t1-3/2. Is the same true for the solut ions of NLKG? Under cer-

ta in circunstances, yes.

Theorem 4.1 (Low-Energy Scat ter ing). Take n=3. Let IF(u) l = O(lul 3) as u ÷ O. Let

u_(x , t ) be any s u f f i c i e n t l y small free so lu t ion (or else assume a s u f f i c i e n t l y small

143

coupling constant) of f in i te energy satisfying

max ]u_(x , t ) I = O(I t1-3/2) (3) x

Then there exists a unique solut ion u (x , t ) of NLKG and a unique free solut ion

u+(x , t ) , both of which are also small, which sa t i s fy (1). The 3 solut ions u_,u,u+

are related by the Yang-Feldman equations.

Le t ~ be the space of i n i t i a l data ~_(0) of solut ions u_(x , t ) sa t i s fy ing

~_(O) G ~ and (3). Then the scat ter ing operator S, which takes ~_(0) in to ~+(0),

maps a neighborhood of the zero solut ions in z in to z.

Examples: This theorem is applicable to F(u) = ± u p with p ~ 3. Also F(u) =

sin u - u. The conclusion has been proved false i f 1<p<1+2/3.

The proof of Theorem 4.1 is not d i f f i c u l t (see /81). I t is based on an i te ra -

t ion procedure (Picard method or Born approximation) fami l i a r to both physic ists

and mathematicians.

The inverse scat ter inq problem is to determine the in teract ion F from the

scat ter ing operator S. Under the condit ions of Theorem 4.1, th is can be done. In

fac t , we define

I ( uv t - utv)dx. B(u,v)

We calculate

d B(u,v) = "(uF(v) - F(u)v)dx dt

Hence l (

B(u+,v+)-B(u_,v_) = l ) (uF(v ) - F(u)v)dx dt (4)

The l e f t side of (4) is determined by S and a pair of a rb i t ra ry inputs u_,v_. Let

u and v are small, say u = E~ and v : E~. Then the r i gh t side of (4) can be ex' -

panded in powers of ~. I f F is a power series (analy t ic func t ion) , then each coef-

f i c i e n t can be successively determined. Even i f . F depends on x, i t can be determi-

ned. For instance i f F(x,u) = V(x)u 3, then the "potent ia l " V(x) can be recovered

from S(see 118,22,231).

Theorem 4.1 is l imi ted to the consideration of small solut ions. But large ones

are phys ica l ly more in teres t ing . We already saw in chapter I I some of the d i f f i c u l -

t ies which can occur when p is large.

Theorem 4.2 Let F(u) = u 3, n = 3. Then there is another space~ P such that

144

and S: ~ ÷ S

Every so lu t ion u (x , t ) of NLKG with i n i t i a l data in s also decays as in (3).

The proof of th is theorem is long and mathematically t r i cky . But i t s idea is

simple. We go back to equation (8) of chapter I I I whence we observe that

IIo° x dt < ~ (5)

This estimate comes from in tegrat ing (8) over a l l time and bounding the f i r s t term

by the energy. We may assume u (x , t ) vanishes outside a l i g h t cone Ixl > t + k and

thus

I I f ( t ) d t / t < ~ where f ( t ) = u4dx

This is an extremely weak statement of decay. Because t -1 is not i n t eg rab le , f ( t )

could not be a constant and in fact _ ~ i f ( t ) d t is a r b i t r a r i l y small on a r b i t r a r i l y

long time in terva ls I . The proof fo decay continues l i ke a jacking-up process. The

succeeding steps are that : u (x , t ) is a r b i t r a r i l y small on a r b i t r a r i l y long time

in te rva ls ; u (x , t ) ÷ 0 uni formly as t ÷ ~; sup l u ( x , t ) l 2 is integrable, and f i n a l l y

l u ( x , t ) I = 0( t -3/21. x

The most in te res t ing step is the uniform convergence to zero. Let ~ be a posi-

t i ve number. Let T = T(~) be s u f f i c i e n t l y large. By the preceding step, l u ( x , t ) l <

on some time in terva l I t * - T , t * I. Let

t** = s u p { s l lul < ~ in I t * - T , s l }

I f t * * = ~, there is nothing to prove. Suppose t * * < ~. Take a time t s l i g h t l y

l a te r than t * * ; namely t * * ~ t ~ t * * + ~. Break up the r i g h t side of (2) in to four

parts. Since t ~ t * * ~ T is large enough, lUol < ~/4. The integral over I t * * , t l ,

the t i p of the cone, is less than E/4 i f a is chosen small enough. In the in terva l

I t - T , t * * I , we have l u ( x , t ) I < c. Since u appears in (2) to the th i rd power and

is small, we can arrange the integral over I t - T , t * * I to be less than c/4, no matter

how large T is . The fourth part is over the large base of the cone lO , t -T l , where

we do not know that u is small. However t - s > T in that in terva l and so

R ( x - y , t - s ) is small in some sense. The kernel is ac tua l l y constant on the hyper-

boloids ~ = constant, but they bunch together very c losely and contr ibute l i t t l e to

the in tegra l . Al together, we obtain l u ( x , t ) l < 4(~/4) = c,which contradicts the de-

f i n i t i o n of t * * . This proves the uniform decay. For the deta i ls of th is proof, see

1231.

One can prove the fo l lowing propert ies 1231 enjoyed by S.

145

(a) S maps~ C one-one onto S

(b) S is a diffeomorphism on S

(c) S is Lorentz- invar iant .

(d) S commutes with the free group exp i t H o-

(e) S is odd,

( f ) IISfll = l l f l l (energy norm)

(g) S is not a l i near operator

Theorem 4.1 is rather eas i ly generalized. I t has the same character as do Theo

rem 2.2 and Klainerman's new resu l t mentioned at the end of chapter I I , Theorem 4.2

is more del icate. I t generalizes to ( i ) the mass-zero case s t i l l with n=3 and to

( i i ) NLS in any dimension. Actua l ly these two cases are considerably easier than

NLKG because of the conservation laws: equation (7) of chapter I I I for ( i ) and equa

t ion (11) of chapter I I I for ( i i ) . In e i ther case we have

I G(u)dx = O(t -2)

which is a far stronger estimate than (5). In the case of NLS the analogue of Theo-

rem 4.2 is va l id i f F(u) = lu lP- lu i f

l + 4 / n ~ p < I + 4 / ( n - 2 )

These two special values of p are fami l ia r form our eas l ie r discussions. This re-

su l t was proved in the elegant papers of Ginibre and Velo 1211 using the conserva-

t ion law which they discovered. For n=3 i t was also proved by Lin and Strauss

[J. Funct. Anal, 1978] who generalized the method used for NLKG.

IV.3 Scattering of a rb i t ra ry f i n i t e -ene rgy s o lut igns

In standard l inear scat ter ing theory the tota l Hamiltonian generates a un i tary

group of operators and so S too is a un i tary operator. I f S is defined on a dense

set in ~L , i t automatical ly extends to a l l o f ~ . In our nonlinear problems S is

defined on a dense s u b s e t ~ of the H i lber t space~-. Is i t possible that S map

a l l of ~ in to~C?. A par t ia l resu l t in th is d i rect ion is the fo l lowing.

Theorem 4.3 Consider NLKG with m > 0 in any dimension n. Assume F(u) = lu lP- lu

where

1 + 4/n ~ p ~ 1 + 4 / ( n - i )

146

Let u_(x, t ) be any free solut ion of f i n i t e energy. (a) Then there exists a solution

u(x , t ) of NLKG such that f lu(t) - u_(t) II ÷ 0 as t ÷ - ~ (b) I f llu_(O)ll is s u f f i -

c ien t l y small, then there exists a free solut ion u+(x, t) such that

f lu(t) - u+(t) l l ÷ 0 as t ÷ + ~. (c) I f --]IlulP+idxdt is f i n i t e , then n+(x, t) ex is ts .

We conjecture that the hypothesis in (c) is always true. I f that were so, the

scatter ing operator S would be a mapping of a l l o__f ~I~ into ~ .

The proof of Theorem 4.3 is based on the funct ional -analy t ica l techniques of

Ginibre and Velo and on some new a pr io r i decay estimates for the free KG equation.

We now state these estimates.

Consider a free solut ion

v t t - Av + m2v = 0

v ( x , O ) = o ,

, m > O, xGIR n

v t ( x , 0 ) = g ( x )

I ts energy is f i n i t e i f gGL 2. Consider the mapping g ÷ v. The f i r s t estimate 1241

is

I v ( x , t ) l q dxdt ~ const (Ig2dx) q/2 (6)

i f 2+4/n ~ q ~ 2+6/(n-2) (q < ~ i f n=1 ,2) . In par t i cu la r , a l l f in i te-energy solu

t ions decay in some sense as I t I ÷ ~. The second estimate 1251 is

I const i lq' 1 I v ( x , t ) l q dxdt ~ t2 ( I g ( x ) dx) q-

where q' = q/(q-1) and 2+4/n ~ q ~ 2+4(n-1) . In order to prove them, one makes a

Fourier decomposition

v ( x , t ) = c le ix 'k (m 2+k2) -1/2 sin t(m 2+k 2 1/2 ~(k)dk

and then uses techniques of Fourier analysis and the interpolat ion theory of opera-

tors.

Theorem 4.3 uses these estimates with q = p + l . Part (a) is proved in 1261. I f

llu_(O)ll is small, then fSlu_IP+ldxdt is also small by (6). Using an i te ra t ion pro-

cedure, we deduce that S~IulP +I dxdt is f i n i t e . Thus (b) is reduced to (c). Part (c)

is proved by the techniques of 1261 although the proof is not given there. As for

our conjecture, there is considerable evidence to support i t : see estimate (5).

147

IV.4 Inverse Scattering

We have already mentioned the nonl inear inverse scat ter ing problem. One of the

most exc i t i ng developments in recent years in nonl inear par t ia l d i f f e r e n t i a l equa-

t ions is the theory of so l i tons 1271. The key discovery was that certain h igh ly

nonl inear problems can be reduced to l inear ones, in pa r t i cu la r to the inverse scat-

ter ing problem for the l i near Schr~dinger equation. I ts main l im i t a t i on at present

is that the technique is successful only in one dimension. The tool used by every-

one has been the celebrated Gelfand-Levitan equation.

Very recent ly Dei f t and Trubowitz 1281 have formulated a new approach to th is

problem which provides ins igh t in to i t s structure and hope of i t s extension to

higher dimensions. They consider the Hamiltonian

H = - d2/dx 2 + V(x) , - ~ < x <

there are two points at i n f i n i t y (±=) so that the scat ter ing operator is a 2x2 ma-

t r i x in t h i s case. Let f+ and f_ be the solut ions of Hf = k2f with f± ~ exp(±ikx)

as x + ± ~. One eas i ly shows that there are functions T,R+ and R_ such that

T(k) f±(x ,k) ~ e ±ikx + R±(k) e $ikx

as x ÷ ~ ~. Then S turns out to be the matrix

~T(k) R+(k)~

S(k) : \R_(k) T ( k ) /

Let us take the case when H has no bound states. The problem is to recover the po-

ten t ia l V(x) from the re f lec t ion coe f f i c i en t R+(k). Assume V(x) ÷ 0 s u f f i c i e n t l y

fast as Ixl ÷ ~. The key new idea is the formula

f ~ 2 V(x) = 2_i_i k R+(k) f+(x,k)dk IT - - co

(7)

They write the differential equation in the form

- f ~ ( x , l ) + V(x) f + ( x , l ) : 12 f+(x , l )

wi th V(x) replaced by (7). This is an i n f i n i t e system of ordinary d i f f e r e n t i a l equa

t ions which are coupled through a l l the frequencies k in the in tegra l . They solve

i t by simple i t e ra t i on for f+(x,k) and recover V(x) by formula (7).

148

19 ]1o

REFERENCES

111R.Y. Chias, E. Garmire and C.H. Townes, Phys. Rev. Lett. 13 (1964), 479-482.

121P.L. Kelley, Phys. Rev. Lett. 15 (1965), 1005-1012; V.I. Talanov, JETP Lett. ~ (19~) , 138-141.

131J.B. Bail lon, T. Cazenave and M. Figueira, C.R. Acad. Sci. 28__44 (1977), 869-872.

]41W.A. Strauss, Anais Acad. Brasil. Ciencias 42 (1970), 645-651.

151W.A. Strauss, Comm. Math. Phys. 55 (1977), 149-162.

161W.A. Strauss and L. V~zquez, to appear.

171 H. Berestycki and P.L. Lions, to appear.

18 W.A. Strauss, in Invariant Wave Equations, Erice 1977, Springer Lect.Notes in Physics no.73, p.197-249.

W.A. Strauss and L. V~zquez, J. Comp. Phys. 28, (1978), 271-278.

M. Reed, Abstract Nonlinear Wave Equations, Springer Lect. Notes in Math. No. 507, 1976.

111 R.T. Glassey and W.A. Strauss, Conservation laws for the classical Maxwell-Di- rac and Klein-Gordon-Dirac equation, J. Math. Phys., to appear.

12 L. Gross, Comm. Pure Appl. Math. 19 (1966), 1-15.

13 I.E. Segal, The Cauchy problem for the Yang-Mills equations, to appear.

14 J. Hughes, T. Kato and J. Marsden, Arch. Rat. Mech. Anal. 63 (1977), 273-294.

151J.M. Chadam, J. Funct. Anal. 13 (1973), 173-184; R.T. Glassey and J.M. Chadam, P--roc. A.M.S. 43 (1974), 373-378.

161S. Klainerman, Global existence for nonlinear wave equations, Ph.D. disserta- t ion, New York Univ. 1978.

171G.H. Derrick, J. Math. Phys. 5 (1964), 1252.

181C.S. Morawetz, Comm. Pure Appl. Math. 15 (1962), 349-362; C.S. Morawetz, Notes on Time Decay and S--cattering for some Hyperbolic Problems, Soc. Industr. Appl. Math. Philadelphia, 1975.

1.191C.S. Morawetz, Proc. Roy. Soc. A306 (1968), 291-296.

]201R.T. Glassy and W.A. Strauss, Decay of Classical Yang-Mills f ie lds, Comm. Math. Phys.; also, Decay of a Yang-Mills f ie ld coupled to a scalar f i e l d , to appear.

1211 J. Ginibre and G. Velo, On a class of nonlinear SchrSdinger equations, to appear.

122] W.A. Strauss, in Scattering Theory in Math. Phys. Reidel Publ. Co., Dordrecht, 1974, 53-78.

1231C.S. Morawetz and W.A. Strauss, Comm. Pure Appl. ~lath. 2_55 (1972), 1-31 and 26 (1973), 47-54.

149

124

125

126

[27

128

I.E. Segal, Adv. Math. 22 (1976), 305-311; R.S. St r ichar tz , Duke MatCh. J. 44 (1977), 705-714.

B. Marshall, W. Strauss and S. Wainger, to appear.

W.A. Strauss, in Nonlinear Evolution Equations, North-Holland (1978), to appear.

A.C. Scott, F.Y.F. Chu, and D.W. McLaughlin, Proc. IEEE 61 (1973), 1443-1483.

P. Dei f t and E. Trabowitz, Comm. Pure Appl. Math., to appear.

This work was supported in part by NSF Grants MCS 75-08827 and MCS 78-03567.

NON-LINEAR TRANSPORT EQUATIONS :

PROPERTIES DEDUCED THROUGH TRANSFORMATION GROUPS

J. GUTIERREZ m, A. MUNIER m~, J.R. BURGAN,

M.R. FEIX, E. FIJALKOW ~

CRPE/CNRS, UNIVERSITE D'ORLEANS (FRANCE)

ABSTRACT :

Transport equations in configuration space ( l inear and non-l inear

heat equations) and in phase space (Vlasov-Poisson systems for plasmas,

beams and grav i ta t inq gases) are considered in the frame of transformation

group techniques. Both se l f - s im i l a r and more qeneral groups are introduced

to f ind spec ia l ly interest ing solut ions. Two kinds of results are obtained :

time evolution of given i n i t i a l s i tuat ions and systematic derivat ion of

possible scaling laws for a given mathematical model. These las t results

are spec ia l ly in terest ing for extrapolat ing performances of Fusion Machines.

Supported by C.I.E.S. France. Permanent address Universidad

de A1cala de Henares Madrid (SPAIN).

Compagnie Internat ionale de Services en Informatique Paris.

U.E.R. Sciences

152

I . INTRODUCTION

In the las t few years, non-l inear problems have become

of major in teres t in theoret ical and experimental Plasma Physics as

in other s c i e n t i f i c areas. Nevertheless, a f te r much hard work by

several authors who have developped a number of analyt ical ~1)" " (2) (3)

and numerical (4} (5) methods to solve non-l inear plasma equations,

only a few quasi - l inear problems have been invest igated.

We intend to present here some ideas which can suggest new

ways to study d i f fe ren t phenomena, not only in Plasma Physics, but in

other branches of Mathematical Physics.

F i r s t l y we shall look at the applications of transformation

group theory to the solution of non-l inear par t ia l d i f f e ren t i a l equations.

This technique was propounded by Sophus Lie (6} at the end of the nine-

teenth century and i t has been applied in the las t decade (7) (8) (9) ( I0)

We sketch quickly th is method ; consider a non-l inear

(or l inear) par t ia l d i f f e ren t i a l system of equations with n independent

var iables. Find a group of transformation which keeps th is system formal ly

invar iant ~. Form those par t icu lar combinations of the independent and

dependent variables which stay invar iant under the transformation, and

cal l them the " invar ian ts" . Now, any function which depends on the

" invar iants" only, is obviously invar iant i t s e l f . Then, look for par t icu lar

solutions of the system, as par t icu lar functions of the invar iants . After

subst i tut ion into the PDEs obtain a new system of equations that these

par t icu lar solutions must sa t i s f y . This system is the '~educed system"

and usually contains less than n new independent var iables. Evidently,

due to th is smaller number of independent var iables, the new system is

often easier to solve and in some cases analyt ical solutions can be found,

but i t w i l l o rd ina r i l y be also non-l inear. On the other hand, i f the

problem is time dependent and i f i t is possible to "el iminate" ( i t would

be more correct to say to absorb) the t - va r iab le , the solutions of the

reduced system involve the whole temporal behaviour of the corresponding

i n i t i a l conditions.

( i . e . the equations remain ident ical except that new variables have

replaced the old ones)

153

For the time dependent problems most of the solutions

obtained by th is method correspond to unphysical i n i t i a l conditions.

A s imi lar d i f f i c u l t y is found for stat ionary problems in re la t ion

to boundary conditions. However, for some cases one can obtain the

analyt ical and tota l time behaviour of some subsets of the physical system (11) .

Our philosophy consists in studying thoroughly a problem

from the analyt ical point of view, in order to s impl i fy i t , using the

transformation group technique as far as possible, and to return to

numerical methods i f we are compelled to by the nature of the solut ion.

To sum up, we give p r i o r i t y to the analyt ical methods and

often we work in a double analyt ical and numerical frame.

In paragraph I I the implementation rules of the transformation

group technique are explained together with the connection between se l f -

s imi la r and in f in i tes imal groups, with the non-l inear heat equation as

an example.

We are mainly interested in phase space f l u ids , which are

defined in Section I I I , where we give some comments about t he i r invar iant

solut ions. Another important application of se l f - s im i l a r groups is shown

in IV : I t is the poss ib i l i t y to obtain scaling laws for experimental

devices (Fusion machines in th is case). We are not interested in the

invar iant solutions of th is problem, but simply in the group invariants

which w i l l give the scaling between technological variables wherever

possible (Here we w i l l "project" the experimental knowledge of one machine

to build a " fami ly" of new improved devices).

Due to the d i f f i c u l t y of f inding physical ly invar iant solut ions,

the idea of quasi-invariance transformation groups is introduced in para-

graphs V and VI. The technique consists in s impl i fy ing the problem as

much as possible from the analyt ical point of view. The objective is the

renormalization of time and forces in order to s impl i fy numerical calcu-

lat ions and obtain information on asymptotic behaviour. We shall give

the connection between these transformations and those of analyt ical

mechanics.

154

I t must be said that the existence of the above group of

transformation (invariance and quasi-invariance groups) is not always

possible and other groups of transformations ought to be invest igated.

I I . TRANSFORMATION GROUPS AND FORMAL INVARIANCE

a) Inf in i tes imal group technique

We are going to give some tedious but necessary technical

deta i ls and precise the notat ion.

Let L = O, a system of par t ia l d i f f e ren t i a l equations in

pr inc ip le non-l inear. The Ul . . .u m, are the dependent variables and

the X l . . . x n the independent ones.

An in f in i tes imal group of transformations G, with parameter

E << 1

111

Xl = xi + ~Ai ( x j , uj) + O(E 2)

ui = ui + ~Bi ( x j , uj) + 0(~ 2)

is chosen in order to keep the system formally invar iant to the f i r s t

order in c. I t makes possible the determination of the arb i t ra ry

functions A i , B i i f they ex is t in a non- t r i v ia l form.

Now, we look for those functions of x i and u i which are

invar iant under the general group G or under any subgroup H c G.

i f T g

That is to say, the functions J(x i , ui) which ver i f y

CG

121 Tg J(x i , ui) - J-(xT, u-i) --- J(x-ii, u-i) = J(x i , u i )

155

D

I cd

• ,4 ,'-4 ~ 0

-,4 r~ ~J

~ ~o

o 0 ~ ~

o -~ o

"~ ~ 0 0.0

0 ~ 0 .~ 0

~ o ~ o

156

Let n be the number of independent var iables, there w i l l be

m < n independentinvariants. Evidently an arb i t ra ry function of these

invariants is i t s e l f invar iant .

In order to find invariants, the corresponding group or

subgroup operators must be used. They are defined in the form

131 Xtg = %i (Ai ~ + Bi Bu i

where A i and B i are the generators of the corresponding subgroup.

In th is representation an invar iant function w i l l be

XTg J(xi ' ui) - J(xTi, u-T) = J(xi, ui)

Obeying in practice the fol lowing par t ia l d i f f e ren t i a l equations

~J ~-~--)= 0 141 zi(A i ~ + B i ~u i

The way to obtain invar iant solutions consists in assuming

that the invar iants are the character is t ics of Eq. 4. These solutions

generate the whole manifold remaining invar iant under the transforma-

t ions Tg (Fig. i ) .

I f these invar iant curves of the manifold are solutions

of the system of par t ia l d i f f e ren t i a l equations, they must sa t i s fy i t .

I t is in th is way that the reduced system is found since the knowledge

of a set of invar iants allows a decrease of the rank of the system.

b) Example : The non-linear heat diffusion equation

Let us start with the one dimensional heat equation with two independent variables x and t

157

[51 !~t u(x, t ) = k-~x I f(u) -~x u(x, t ) ]

where f(u) takes the form

161 f (u ) : u s

The following functional transformation

u 171 ~ : F(U) : I f(u) du

o

allows us to wri te 151 in the form

k H(~) ~2~ 181 a-~: ax 2

except for s =-1, with

191 H(-~)= [ ( s+ l ) u l s/(s+l)

Taking the inf ini tesimal group transformation

2 : u + eB(u,x,t) + e( )

11oi { : t + ET(u,x,t) + e(s 2)

= x + eA(u,x,t) + e(e 2)

158

and invok ing the formal invar iance of equation 181

~-u k ' u s / ( s+ l ) @2~ ~ ,~ s / ( s+ l ) @2~ ~-~ - @x 2 = ~-~ - k @~2

where k' : k (s+ l ) s / ( s+ l ) we obtain

: 0

1111 T = at + b ; A = cx + d ; B = (2c - a) u

The non-essent ia l parameter a in 1111 can be eleminated to give

1121 T = t + b' ; A = c ' x + d' ; B = (2c' - 1) u

These generators give the f o l l ow ing group operator

1131 X : ( t + b ' ) + ( c ' x ÷ d ' ) + (2c' - 1) u B~

and the corresponding i nva r i an t s

X + d ' / c ' u 1141 Jl = ( t ' + b') c' ; J2 = ( t + b ' ) (2c' - 1)

We shal l re tu rn to t h i s problem l a t e r on in connection w i th

s e l f - s i m i l a r methods, and i n v a r i a n t so lu t ions w i l l be obtained. But i t

is very i n t e r e s t i n g to consider the element o f the group given by

b' = d' = O, c' = I , the operator o f which is

115] x = t + x + u ~-6

159

Note that 1151 generates a stretching (homogeneous) group because

i t has the form

n-m i i ~ m bk uk a x • +

i =1 ~x 1 k : l ~u k

where a i" and b k are a rb i t ra ry constants.

This means that the concept of invar ian t solut ions under

a Lie group contains, in part , the concept of the s e l f - s i m i l a r solut ions

or automodel so lut ions, i f they ex is t .

Ord inar i l y the concept of an automodel so lut ion is presented

connected to dimensional analysis, although here th is concept loses

i t s meaning. We are going to apply th is se l f - s im i l a r method to a concrete

example in paragraph IV.

c) Se l f - s im i la r technique

The se l f - s im i l a r groups are one parameter groups, most of

them stretching groups, which allow invar ian t solut ions to be found

for a system of non- l inear par t ia l d i f f e ren t i a l equations. But the

technique to obtain the invar iants is much simpler.

In th is case we take a transformation of the form

1161

xi = a x i

ui = a u i

Let L = 0 be again a system of par t ia l d i f f e ren t i a l equations.

A se l f - s im i l a r group as II61 is applied to keep the system formal ly

invar ian t and th is invariance produces the re la t ions invo lv ing the ar-

b i t r a r y parameters ~i and B i which determine a pa r t i cu la r element of

160

the group. The subsequent analysis of the transformations w i l l give

the corresponding invariants. From this point the same method des-

cribed for inf initesimal groups is used to obtain the reduced system

determining the invariant solutions.

We treat by this method the previous example, the non-linear

heat diffusion equation, and some solutions w i l l be given.

d) Example : heat equation

Taking the heat equation as 181

~-~-u= k' ~n B2~ wi th 1171 @t ~x 2

s n = s---~- T

we invoke formal invar iance a f t e r app l i ca t i on of a s i m i l a r i t y group

= a ~ ; ~c = aBt, ~ : aYx

and we obtain

1

with y and ~ arbitrary constants.

For sake of simpl ici ty one can take

y = w~

to write

1181 ~ = a(2m-1) B/~u ; t = a~t ; R = amBx

161

wi th i nva r i an t s

I191 J l = x (~)-m ; J2 : ~ (~)( l -2m) I /n

We dispose of an a r b i t r a r y parameter m. T is a c h a r a c t e r i s t i c time

f o r the evo lu t ion of the system and the o r i g i n o f time should fo r

convenience be taken at t = T. Since fo r t = T, the i nva r ian ts become

J1 = x ; J2 = ~

(o f course a t r ans l a t i on of time is always possible and we could

in t roduce the time var iab le t -T ) .

In order to obtain i n va r i an t so lu t i ons , we assume

1201 J2 : G(J1)

and we subs t i t u te 1191 in eq. 1171 which becomes

121J 1 dG G n d2G (2m - 1) G(~) - m ~ = Tk'

wi th

n = J1 = x (~)-m

We choose m in such away tha t the l e f t hand side of 1211 m u l t i p l i e d by

G -n is a to ta l d i f f e r e n t i a l . I t gives m = (s + 2) -1 . Then, obv ious ly

except f o r s = -2, eq. 1181 can be w r i t t en in the form

162

d2G 1221 dn 2

= k" ~q (riG I / ( s + I ) )

where

(s + 1) (s + 1) I / ( s + 1) 1231 k" = - k"T (s + 2) = - KT (s + 2)

The inva r ian t so lu t ions obeying the r e s t r i c t i o n

dG q d--~ = 0 = 0 wi th G = 0 a r b i t r a r y

is def ined by the fami ly of curves

G : (an 2 + b) c

wi th b a r b i t r a r y constant (G n = 0 = bc) ' ~ and c def ined by

a = - s(s + I ) - s / ( s + i )

2kT (s + 2)

s + l ; c - s

and tak ing the inverse t ransformat ions we obtain the fo l lowing fami ly

of so lu t ions to the non- l inear par t ia l d i f f e r e n t i a l heat equation ~12)" "

u : [(s + l )u ] I / ( s + I)

1241 l Ax + B(~) 2/(s + 2) ]1 /s

u = ( t /T )

where B is a r b i t r a r y and A is given by

163

, . ~ X ~

I

+~

II

~,,

I I

o ~

o ~ ~ 1 ~

II II

~ .~

A = 2kT (s + 2)

164

In th is case we have a number of in te res t ing physical i n i t i a l

condit ions for instance for a l l values of s wi th s > O. Notice that the

solut ion is truncated below the points where u cancels ( f i g . 2).

I I I . PHASE SPACE FLUIDS

We are going now to consider some solut ions obtained for

those systems, in which we are mainly in terested, namely the phase

space f l u i ds . We ca l l phase space f l u i d , a N-body system described

by a p robab i l i t y densi ty obeying a con t inu i t y equation and other

physical constraints defined on R 2n (conf igurat ion and ve loc i t y

spaces). In con t rad is t inc t ion some f l u ids can be described by the

two f i r s t moments of the p robab i l i t y density which is a considerable

s imp l i f i ca t i on . This is the case fo r water, or non raref ied neutral gas.

a) Three pa r t i cu la r f l u i ds

There are three pa r t i cu la r cases which can be described mathe-

mat ica l ly by the same system of equations with non - l i nea r i t y determined

by an in tegra l term.

The only in teract ions between par t ic les considered in the

system w i l l be long ranges ; i ne l as t i c co l l i s i ons and short range elas-

t i c co l l i s i ons are neglected because they are exceptional processes,

or because the evolut ion of the system is faster than the time between

two close processes at a short range

1 T co l l i s i on less << c o l l i s l o n

where v c o l l i s i o n is the c o l l i s i o n frequency and T co l l i s i on less the

charac te r i s t i c evolut ion time of the system when only long range

processes are considered.

165

These three co l l i s ion less phase space f lu ids are : two

species plasmas, one species beams of charged par t ic les and s t e l l a r

systems.

A two species plasma is a system of two kinds of par t ic les

with opposite charge (ions and electrons) interact ing accordingly to

Coulomb's law.

A one species beam is a system of ident ical charged par t ic les

(ions or electrons) where the Coulomb forces w i l l be only repulsive.

System constituted by N massive bodies ( i n t e r s t e l l a r gas,

set of stars or galaxies, e t c . . . ) interact ing between them via the

grav i ta t ional forces w i l l be called s t e l l a r se l f gravi tat ing systems.

These co l l i s ion less f lu ids w i l l obey L iouv i l l e ' s equation

with the i r corresponding electromagnetic or gravi tat ional Hamiltonians.

In the charged par t ic le case the poss ib i l i t y of self-induced electro-

magnetic radiat ion f ie lds must be taken into account due to the mobi l i ty

of par t ic les.

b) Mathematical description

Due to the imposs ib i l i t y of solving Liouvi l le~s equation in

R 2n, s impl i f ica t ions must be introduced in the i n i t i a l problem. For

the above mentioned cases the BBGKY hierarchy (13) can be used with

addit ional assumptions about the d i f fe ren t correlat ion functions.

Likewise we can fol low the Rosenbluth and Rostoker method (14) to reach

the same resu l t as

Vlasov + Maxwell equations -

In addit ion, for the plasma, the extrapolat ion to i n f i n i t y

for ionic masses is usually considered because they have very slow

dynamics compared with electrons. In th is way only an evolution equa-

t ion for one species of par t ic les remains, with ions appearing as a

constant density in the divergence of the e lec t r i c f i e l d equation

(see Eq. 1291).

166

To sum up, the Vlasov approach is equivalent to take the

l i m i t of the mass and charge of electrons as going to zero, with the

density of par t ic les going to i n f i n i t y such that ~, ne and nm remain

constant (e, m and n e lec t ron ic charge, mass and density respect ive ly) .

In th is l i m i t the plasma becomes a continous f l u i d .

This approximation can be v isual ized assuming that we "cut"

the physical par t ic les in two ident ica l parts. Each new "par t i c le " is

again cut in two parts and so on, obtaining f i n a l l y a "mash" which

represents mathematically a continous f l u i d wi th loss of the grain

character of the i n i t i a l descr ipt ion.

The determining parameter of th is s i tua t ion is defined

in the form

1 1251 g = 3

nL D

where L D

a plasma

is Debye's length or range of the screening e f fec t . For

~o kT I /2 1261 k D = ( ~ )

47 ne

k being the Boltzmann constant.

Non-col l is ional cases correspond to

g << 1

whereas g > i indicates that ind iv idua l ef fects are at least of the

same order of magnitude than co l lec t i ve ones.

167

g ÷ 0

The equations obtained for plasma f l u ids are in the l i m i t

271 ~f ~f ÷ ~-~ + ~ .3 -~ + y . ~ : o

÷ e (~ + 1 ~ x ~) 2.81 y : ~

291 d iv~ : 4 7 e [ f f d 3 ; - ( N o ) ] - oo

+ 47 Pext

301 div ~ = 0

311 rot ~ = c @t

321 rot ~ - 47 ~ + I dE C C 3 t

where N o is the constant density of ions, Pext corresponds to the

external charges creating the external f i e l d , the integral term in

eq. 1291 is the mean f i e l d produced by a l l electrons in the plasma,

and f the p robab i l i t y density in phase space, being a funct ion of

r , ~ and t

f = f ( r , v , t )

The mathematical descr ipt ion of a Beam is ident ica l to the

above one except in the equation 1291 because N o is here equal to zero.

In these two cases we have wr i t ten the most general system

of equations. Neverthless, there ex is t in teres t ing and pract ical l im i t i ng

168

cases cal led "e lec t ros ta t i c " where the self- induced magnetic f i e l d due

to mobi l i ty of par t ic les is neglected. The equations are reduced to

the Vlasov-Poisson system (moreover we have also taken the external

magnetic f i e l d equal to zero)

1331 mf + ; . ~ f + e ~ . ~ f : 0 ~t ~F m ~;

~ - 41Tel f f d3~- (No)] 1341 ~;

with N o = 0 for the beam case. The other Maxwell equations are

automatically sa t is f ied .

For a gravi tat ional system the equations are formally

ident ical to the beam case ones when the external forces are not

considered. The only di f ference is the change of sign for the f i e l d

divergence because the a t t rac t ive character of interact ions between

par t ic les . Equations 1331 and 1341 become

1351 ~f @# + ~ . ~f - 0

1361 div ~ = -4~ G o m j f d3~

The techniques pointed out in the previous paragraph have

made i t possible to f ind some singular solutions for these three phase

space f lu ids . These solutions exh ib i t d i f f i c u l t i e s on boundaries with,

sometimes, par t ic les located at i n f i n i t y in configuration space and

i n f i n i t e k inet ic energy. Nevertheless a physical meaning can be given

to these analyt ical solutions (11). The used concept is cal led

contamination and is based on the v i r tua l separation of the par t ic les

in two populations, a central one (which has a physical meaning),

and the external par t ic les (which in fact we want suppress). The

169

The crucial point is that for systems with a su f f i c ien t degree of

symmetry the interact ions of the external par t ic les on the central

populations manifest i t s e l f only a f ter a certain time in the worst

case and sometimes never manifest, at least for some subsets of the

central (physical ly meaningful) population. This concept has been

quite useful especial ly in the beam case.

Applying the se l f - s im i l a r group techniques to equations 1331

and 1341, we get the fol lowing invariants

1371 s = f(_~)~+z ~ = ~(T )2-~ ~ ~-- ~(T )-~ ~ ~ = ~V(y)t1-~

with m a real a rb i t ra ry parameter and T the character is t ic time in

the evolution of the system. We obtain, sust i tu t ing 1371 in 1331 and

1341 for a one dimensional system

aF e aF I I< ii 1 - ~ ~ + ~ ~ + (~+1) F

-Foo

~ - 4~Te[ / Fdq- NO(-~)2 ] 1391 ~ - c o

= 0

Again, the time or ig in must be taken at t = T. Unfortunately, for

N o # 0 the variable t is s t i l l present in the f i e l d divergence equation,

which makes i t impossible for the stretching groups to be used in the

plasma case. But, for the gravi tat ional and beam cases, N o = 0 invar iant

solutions can be obtained from eqs. 1381 and 1391. In the one dimensional

simple Water Bag model, open invar iant solutions have been found (15)

(e.g. solutions with par t ic le at i n f i n i t y in configuration space).

However, as we have already pointed out, these solutions give in for -

mation for a l l times on the behaviour of the central part of a physical

system with closed i n i t i a l conditions. The proportion described

ana ly t i ca l l y depends on the i n i t i a l dimensions of the system in phase

space. These solutions are called "rod" solutions because of the i r

form (see f ig . 3).

170

G V

i L • i I i

b v

FIGURE 3

(3 V

i i i

Self-similar solutions of a beam of identical changed particles in a

single Water-Bag model, a) initial condition ; b) temporal evolution.

The density decreases with time.

FIGURE 4

Collapse of the self-similar solutions of a gravitational problem in

a single Water-Bag model, a) initial condition ; b) temporal evolution

after a finite time.

171

In the one dimensional gravitational case we find solutions

with the same structure, but their temporal evolution induces a genuine

gravitational collapse, the energy being, after a f in i te time,

concentrated in a very l i t t l e region around the center of mass of

the system. The model used is again a simple Water Bag model (see

f ig. 4).

For the non ini t iated reader let us remember the concept of

the simple one dimensional Water Bag model. All particles are i n i t i a l l y

located with the same probability in a closed region of phase space ;

that is to say the distribution function f is constant in this region

and takes a zero value everywhere else. The Vlasov equation preserves

this constant value of f for any time and consequently we have just

to study the time evolution of the boundary curves which represent

the upper, V +, and lower, V', velocities in phase space. They obey

the following equations

@E A(V + V- ) (No) [41l a-~ = -

A is the constant value of the distr ibut ion function. Obviously any distr ibut ion function in phase space can be approached by a multiple Water Bag model, but we are not going to study this generalisation.

For a three dimensional gravitational system with spherical

symmetry the corresponding equations can be simplified to (24)

~F ~F ( V±2 ~ # ) ~F _ 2 V// V~-2 ~F 1421 + v// + r 0

1431 A @ = 4~ GoP

172

with p(t,r) = 11- i dr// / d(V~ 2) F(t,r,V~2,V//) - ~ 0

1441 Dr

G 0

7 m

m(r, t ) = ~T

r

0 -oo O

where F(t,r,V//,V± 2) is the d is t r ibu t ion function of the system sat is -

fy ing the Vlasov (Boltzmann) equation 1421 with the independent

variables : time t , radius r , radial ve loc i ty V// and the square of

tangential ve loc i ty V± 2. @ is the interact ion potential in the system

with p ( r , t ) the par t ic le density and m(r , t ) the mass located within

a distance r at a time t . G o is the grav i ta t ional constant.

In th is example we w i l l apply the concept of se l f - s im i l a r

invariance twice, up to the step where analyt ical resul ts can be

obtained. We hope to s t i l l retain some information about the non

l inear behaviour of the system.

At f i r s t , we "absorb" the time in the new variables and

functions ; the invar iants obtained are

14sl

H = F(t/T) 2-3m ; M = m(~) 3m-I

= r ( t l T ) ~- I ; n = V//(tlT) ~ ; X : V 2(t lT)2~

which generate the fol lowing reduced equations

1 I @H 3H 3H} [461 H + T (~-I) ~-~ + ~ ~ ~-~ + 2 ~ - ~

+ n ~ + ~ x- G o ~ ~ ~-~ =

173

dM 4 ~ 2 / [471 ~ = dn dX H ( ~ , n , X )

- co 0

with T the character is t ic time in the system ( i n i t i a l time corresponds

to t = T), and m an arb i t ra ry constant.

Now we absorb the space variable ~ through a new applicat ion

of the se l f - s im i l a r group of transformations technique. In th is process

the reduced mass M of the system w i l l become a constant. The corres-

ponding invar iants are here :

1481

D : H(~/R) 3 ; ~ = M(~/R) -3

: n(~/R) - I ," @ = X(~/R) -2

giving the equations

R ~D 1491 D( - 3m) + (@ _ 2 + m # _ Go ~) --am

R @D _ + (2 ¢ ~- - 4q~o~) a@ 0

- / 4 R 3 / 1501 ~ = ~ ~ dw dq~ D(~,q~) - co 0

In 1481 and 1491R is the space scaling factor.

At th is step the solutions of eqs. 1491 and 1501 can be

calculated, and we can wri te

, i 0

174

where K(~) is an arb i t ra ry function of i t s argument ~ and the new

variables are in terms of the old ones

p = 4m - 2B

q = 44

1521

B = R/T

A = 4(R) 2 (I - j2 T 2)

Notice that A is a parameter which depends on the Jean's frequency

1531 J = (4~ G o po )1/2

Po being the mean density in the system.

Coming back to the i n i t i a l variables r, t ,V// ,V~ 2 we notice

that th is solut ion corresponds to the expansion of a system with a

density of par t ic les uniform in space and varying as Po(t/T)-2.

The nonl inear i ty of the problem subsits through the fact

that only some values of A are permissible. These values are obtained

by noticing that D must ve r i f y the Poisson equation 1501. In fact

i t w i l l ve r i f y 1501 provided the integral of the l e f t hand side ex is ts .

The l im i t i ng values of A are 4B 2 (with corresponds to a b a l l i s t i c

expansion with a negl ig ib le e f fect of the grav i ta t ional forces) and B 2 j2 T 2 A = ~ (which corresponds to = ~). This solution corresponds

to the slowest se l f s imi la r expansion ( for a given i n i t i a l density po).

175

J~

/I

I I I . I I ~ "4- o'3

I , , I I I

(N ~---

I I I I i , I I I i

f_

t' , , .

, 0

u~

",4-

cO

f_

,,0

uO

',4"

oZ,

O,

-,~ •

"~ 0 ..~

LM

o - - I~1"

o ~

-,-t

0 II ~ I ~-I l l4

• ~ ~ 0

o

176

These solutions present the same d i f f i c u l t y as in the

previously mentionned one-dimensional case, since they have par t ic les at

i n f i n i t y in configuration space. We show in f igure 151 the phase space

structure of some solution of 1511 - Those for which V~ is always zero

- We can appreciate the i r open structures. As in the one-dimensional case

a central part describes an i n i t i a l closed physical system. For th is

central part an analyt ica l descript ion is made possible by the in t ro -

duction of the concept cal led "contamination e f fec t " .

We have given th is rather lengty calculat ion to show

how by successive appl icat ions of the se l f s imi la r technique a very

complex coupled non-linearsystem of equations can be brought to f u l l y

analyt ical analysis. Of course, in th is process many in terest ing

physical s i tuat ions have been suppressed. Nevertheless we s t i l l recover

in terest ing physical resul ts on the expansion of homogeneous systems,

the role of grav i ta t ional forces and the i r ve loc i ty structure.

In the next section we are going to present a very in terest ing

appl icat ion of s e l f - s i m i l a r group techniques where we shall not be

i n te res ted in the inva r i an t solutions but in the invar iants themselves,

because they give us enough information to construct a scaled system

from an i n i t i a l one. Notice that a se l f - s im i l a r transformation is often

a stretching transformation, and th is means a rescaling of the variables.

IV. AN APPLICATION OF SELF-SIMILAR GROUPS : SCALING LAWS

The theory of models can be assimilated to the existence

of some combinations of the physical parameters characterizing a system,

which remain invar iant under a transformation group. Consequently changes

in the parameters of the model can be introduced without changing i t s

physics.

In th is sense, a se l f - s im i l a r group can define scaling laws

for a model characterised by a system of equations, since the i r t ransfor-

mations are wr i t ten in the form

1541 xi = a~i xi

177

In p r inc ip le the new equations obtained can be d i f f e ren t

from the i n i t i a l ones, but, here, in order to deduce the scal ing laws

we w i l l impose the formal invar iance, t he i r form remaining unchanged.

I t introduces a system of l i near algebraic homogeneous equations for

the d i f f e ren t mi" I f th is system has solut ion in addi t ion to the t r i v i a l

so lut ion mi = O, we have a possible scal ing law. In fac t , the number

of independent mi is equal to the number of degrees of freedom, i . e .

the maximum number of parameters in the model which can be changed

independently. I f only one mi is a rb i t ra ry we can change one parameter ;

for two mi we can choose a r b i t r a r i l y at most two parameters, although

not a l l pairs are allowed.

Let us suppose that we know the system of equations which

describes the whole physics of a Fusion Machine. Invariance under a

se l f - s im i l a r group w i l l give the group invar ian ts , through which we

can express any solut ion of the system as an a rb i t ra ry funct ion. I f

th is funct ion remains unchanged the physics in the machine is also

unchanged. This is possible i f the parameters are scaled wh i l s t

preserving the group invar iants . We have then constructed a "se l f -

s imi lar " or an automodel machine.

In th is appl icat ion there is a very important fact ; we

are not obliged to know the e x p l i c i t form of so lut ions. In th is

philosophy the measurement of one machine is automatical ly extra-

polated to other machines and th is " se l f - s im i l a r " behaviour defines

a family or determined type of fusion machine for which predict ion

is possible. Now i t w i l l be very in te res t ing to know i f , by scal ing,

they can reach Lawson's c r i t e r i on

1551 n T Z 1014

where n ( in cm -3) is the plasma density and • ( in s) the confinement

time.

178

I t must be pointed out that th is method has a l o t of

l im i t a t i ons . Of course we do not claim to have solved the fusion

problem because the whole physics of any fusion machine is not

known at present. But i t seems to us in teres t ing to study some

physical models of plasmas which are involved in part by Tokamaks

and other experimental thermonuclear devices, in order to decide i f

they are in teres t ing from th is physical point of view.

On the other hand, the choice of a pa r t i cu la r model (or set

of equations) depends on which aspect of the problem is being considered.

For a fusion machine three aspects can be considered

- the formation of a plasma from a neutral gas by

d issociat ion and ion iza t ion

- the heating of th is plasma

- the confinement of the f u l l y ionized plasma.

The f i r s t phase implies phenomena invo lv ing quantum mechanics

and atomic physics. The second phase leads to considering a set of charged

par t ic les in external and in terna l electromagnetic f ie lds from a very

low temperature to a high temperature (about I08°K) in the frame of

c lassical mechanics. The t h i r d phase involves the evolut ion of th is

populat ion, under c lassical mechanics up to i t s disappearance by

fusion processes.

The f i r s t two phases although important from a pract ical

point of view can be forgotten for our der ivat ion purposes and we

shal l suppose that the plasma is in the confinement step.

As has been stated in section I I I the Vlasov model is only

va l id for a plasma parameter g, (as defined in 1251), much smaller

than 1, and i f ind iv idua l ef fects such as cyclotron rad ia t ion , bremstrah-

lung and co l l i s i ons are neglected. This may be true for the confinement

phase.

Let us consider three sucessive approaches to the model

characterized by the Vlasov-Maxwell equations. As we introduce new

179

physical phenomena into our system, the number of degrees of freedom

wi l l decrease. We wi l l see that for Vlasov's model with self-consistant

electromagnetic f ields we have only one degree of freedom and the

introduction of a new physical restriction makes i t impossible to find

a scaling law : for instance, collisions cannot be introduced. Self-

s imi lar i ty properties for more complicated models must be studied and

the crucial question is the val id i ty of these models. I f we can be sure

of the correctness of one of this model we must obtain the transformation

group and see what freedom is le f t .

Here we start from eqs. 1271 to 1321 which describe the

complete Vlasov-Maxwell model. A f i r s t approach consists in considering

only an external magnetic f ie ld and moreover assuming that the plasma

is at such low density that both the self-consistent electr ic and

magnetic f ields are negligible). The system of equations is reduced

to the Vlasov equation only

[561 ~ f ~f B f ~-~ + ~. ~-- + ( ~ x ~ ) . ~-- - o

where ~ is an external f ie ld which we can arb i t ra r i l y vary.

The self-similar group which leaves invariant eq. 1561 is

157] *r : a ~ ; ~ : a ~ f ; {: : a - ro t

V : a ~ ; : am ~

the corresponding invariants being

158]

v i vi3f J1 = Bit ; J2 = rj(1+c/b) ; J3 = rj(I/b+3c/b+3).

B i J4 =

180

We note that I , m and m in 1571 are three arb i t ra ry

parameters which give the poss ib i l i t y of obtaining a system with

one, two, or three degrees of freedom. In the expression 1581 i and

j define any component of the vectors ~, ~ and ~. Due to the fact

that only three variables are independant one of the four invar iants

w i l l be an arb i t ra ry function of the others. The invar iants are

expressed as functions of the variables characterizing the machine.

We w i l l , a r b i t r a r i l y , select four quant i t ies namely the radius of

the machine "a", the confinement time %, the temperature of the

plasma T and the plasma density n ; then expressions [581 can be

wr i t ten in the form

T n a @, J4 = B a ~ J1 = BT, J2 = a2-~B ' J3 =

with @ and ~ a rb i t ra ry parameters. The determined values of ~ and

define a family of se l f - s im i l a r machines. Choosing J1 as a function of

J2' J3 and J4' we obtain

]59[ B% = F ( a - ~ , n a @, B a ~)

F being an a rb i t ra ry function. I f F is f ixed, a par t icu lar type of

machine is obtained. Now we can separate the devices with d i f fe ren t

degree of freedom

a) For @ = ~ = -~ we have a machine with three degrees

of freedom

16ol B~ = F(a-~B2)

We consider the 4 parameters describing the machine and T which

indicates the performance. Among the 3 parameters T, a and B

181

2 can be chosen a r b i t r a r i l y and moreover n remains always completely

a rb i t ra ry . The las t of the three parameters T, a and B and the perfor-

mance T are subsequently f ixed. We can describe the system as possessing

three degrees of freedom. This is obviously the most in te res t ing and

general scal ing law for th is model where se l f consistent ef fects are

neglected.

b) For @ = -~, ~ > -~ we have a device with two degrees

of freedom

1611 BT = F(a-~- ~ , B a s ) with s f ixed a r b i t r a r i l y

Here the choice of scal ing factor for a var iab le, allows an a rb i t ra ry

choice for two others, although not a l l pairs are permissible.

For each value of s a new se l f - s im i l a r family of machines

is obtained.

c) For ~ > -~, ~ = -~ we obtain again a machine with two

degrees of freedom

162[ B% = F ( a - ~ , n a r)

d) For # > -~, ~ > -~ devices with one degree of freedom

only are obtained

163] BT = F(a-~- ~ , n a r , B a s )

As has been said previously, i f new physical res t r i c t i ons

are considered the a rb i t ra ry parameters # and ~ w i l l be f ixed and

the number of degrees of freedom w i l l decrease.

182

Let us consider now the electrostatic approach to this

problem. The magnetic f ield is external and only the self-consistent

electrostatic f ie ld is taken into account. From all the Maxwell

equations only the Poisson equation is retained, a hypothesis obtained

formally by lett ing c(velocity of l ight) go to in f in i ty in the

Maxwell equations127-321 . The corresponding equations describing the

system wil l be

1641 Bfi @fi @fi + ; . + ( ~ + ; x g ) • - o

+oo

165E ~" ~ : ~ / fi d3v i

- oo

where i indicates the different species in the plasma, that is electrons

and ions. The self-similar group which leaves invariant 1641 and 1651

is defined by i ts transformations

L66l f = a-(3X+co)f ; r = al~ ; { = a-C°t

V = a (I+~) ~ ; B = acoB ; E =

where we have only two arbitrary parameters and therefore at most two

degrees of freedom.

The corresponding invariants can be contructed in a similar

way to the previous case

T : n = a@ J1 = BT ; J2 = ~Lr-~ ; J3 B-~ ; J4 B

with @ an arbitrary parameter.

183

As for the previous case ¢ > -® give a self-similar set

of devices with one degree of freedom and for ¢ = -~ we have the

interesting case of two degrees of freedom machines. The corresponding

function is

1671 B% = F ( a - ~ , B~ )

I t is very easy to prove that i f a coll ision term is added to eq. 1641

to obtain Boltzman~s equation the parameter @ is fixed with the value

@ = 3/2. In that case we can vary arbi t rar i ly only one parameter, all

the others and the performance being consequently deduced. We obtain

1681 BT : F ( a - ~ , B- ~, a3/2B)

To f inish, let us apply the self-similari ty techniques to the

whole Vlasov-Maxwell system defined by the eqs.

f i Bfi Bfi - - + 7 . + ( ~ + V x ~ ) - ~t ~F ~

div~ = ~ / fi d3~ i

- co

d iv~ : 0

rot ~ = - I/C @~

rot B 4~ i 3~ : ~- ~ + T T ~

184

which remain invar iant under the group characterized by the fol lowing

transformations

f i -% a-m t : a-m = a2m f i ; r = g ; t ;

V = V ; B = a ; =

Here we have only one degree of freedom and the corresponding family

of se l f - s im i la r machines w i l l be defined by

I691 BT = F(a- ~ , ~ , B a)

which derive from the invariants

1701 f i

J1 = BK~ ; J2 = VK ; J3 = ~ ; J4 = B K r j

As before K and j mean that the invar iant is defined for any component

of the vectors B, r and V.

I t is not possible now to consider new physical res t r i c t ions

to th is problem, because of the incompat ib i l i ty of the homogeneous

determinant system. For example i t is impossible to get scaling laws

for a f u l l y electromagnetic plasma including ef fects of co l l i s ions .

As mentionned in 1691 using only the e lec t ros ta t i c approximation allows

the inclusion of co l l i s ions . 1681 and 1691 are the two most complete

scaling laws for plasma.

Before commenting the previous results, we must say that

a similar analysis has been done by Connor and Taylor ~17)" " from a

dif ferent point of view, and for part icular cases contained here since

a s l igh t ly more general treatement has been used here. We w i l l consider

185

only scal ing laws as given by 1691 and we note that the temperature T

is an invar ian t . This means that no scal ing law ex is t for machines

at d i f f e ren t temperatures, which turn out to be the most d i f f i c u l t

th ing to extrapolate physicaly. As soon as i t is possible to obtain

the correct temperature we can f ind in teres t ing s i m i l a r i t y laws on

the inf luences of "B", "n" and "a" on %.

But there is one in teres t ing idea that can be developed.

Due to the fact that each funct ion F defines a fami ly of se l f - s im i l a r

machines we can look at the ef fect on the physical variables in the

machine when one of these variables is changed a r b i t r a r i l y and the

others subsequently deduced.

Each machine w i l l now generate a one parameter fami ly

( i . e . having one degree of freedom) defined by a par t i cu la r funct ion F

with the same plasma dynamics. The in teres t ing point is tha t , for th is

fami ly , the fusion performances w i l l be d i f f e ren t and also the techno-

logy. Fusion performances are measured by the product n T of the

density with the confinement time. Let us consider the n % diagram

in Figure 6 and l e t us draw the curve corresponding to the fami ly

generated by a given device. Since logari thmic scales are used,

Lawson's c r i t e r i on

L711 n ~ = 1 0 1 4

is represented by a s t ra igh t l i ne . The e l iminat ion of B between the

invar iants J l and J3 in 1701 gives

2 n % = const

for the equation in n, T of the fami ly. Consequently, from a given

machine we can deduce a family with better fusion performances. To

increase the product n T by a factor ~ we can

186

0 r

I I

E

Z

o

h i

U

Z 0 u

P

/

/ /

I I I I I

,~ o I u u I

I .~ 0- I

,4 v

! C~

0

h i C~

LI. I - -

I

0

I ? 0

I / ,

/ / I I I i I

,I i I I

I I

I

!

,-/ I

/ ' I

I I

I

I

0

I / ,

I 0

I

o

o o

v • m ~.0

• o

D,, ~ .,-,I m

b

• ,4 0

. ~ ~

0 ~ r~ 0

q~ m ~ 0 0 ~

• ,4 ~ 0

.,4 ~.~ ~ ~

• ,~ E~ .ca,~ 0 n~

0 ~.~ ~'~

0

°~

~ - - ~ ~ ~o ~.~

187

i ) increase the magnetic f i e l d by the same factor

i i ) decrease the charac ter is t i c size of the system by

th is factor

i i i ) increase the density by 2

The problem is now technological . Is i t easier to bui ld

smaller machines with increased magnetic f i e l d ?. To specify the

problem we look at d i f f e ren t machines and extrapolate them to the

Lawson c r i t e r i on . TFR implies an increase of the magnetic f i e l d by

more than two orders of magnitude (and the subsequent decrease in

s ize) . This gives a r id icu lous machine of i cm, with a magnetic

f i e l d of 5 x 106 gauss !

"Alcator" extrapolat ion is less unphysical. The ra t i o

to bring i t on the Lawson curve is 12, implying a magnetic f i e l d of

106 gauss with a torus radious of 5 cm. Nevertheless such a machine

would be s t i l l qui te unbalanced and impossible to bu i ld .

Thing change with JET fami ly. Start ing from a projected

= 0.35 s. and n = 5 x 1013 par t ic les per cm 3 (and consequently

n % : 1.5 x 1013 ) we w i l l increase n T to 1014 with a factor of 6

on the magnetic f i e l d (up to 18 x 104 gauss) and a decrease in size

from 3 to 0.5 m. This c lear ly points out the advantage of a techno-

logical solut ion having an intense magnetic f i e l d and suggest that

there is an in te res t in smaller machines.

We can suggest the fo l lowing ideas :

- to test the scal ing laws by bu i ld ing a set of small

machines as s e l f - s i m i l a r as possible,

to look at the family generated by each magnetical ly

confined machine, performances of which in terms of n T w i l l always

increase for smaller machines with a more intense magnetic f i e l d .

I t must be said that th is statement w i l l be considered under the

188

scaling law philosophy and does not mean, of course, that smaller

machines, all other parameters remaining unchanged, have better

performances, but that we can exchange size against magnetic f ield

and s t i l l obtain better performance.

This reinforces the interest in developping high magnetic

f ield technology which wi l l allow a substantial decrease in size

(and consequently in price) of the machines. Connor and Taylor (18)

have arrived at similar conclusions. Of course all the physics taking

place in a Tokamak can hardly be brought down to the Vlasov-Maxwell

model. Nevertheless this last model describes certainly very well

the dynamics of the particles.

V. TRANSFORMATION GROUPS OF PARTIAL INVARIANCE

a) Transformations in time-phase space

As we have mentionned above, the formal invariance of a

system of differential equations under a group of transformations

presents two main di f f icul t ies : not any invariant solution is physical

and solutions can be found only for a subset of in i t ia l or boundary

conditions. In agreement with our philosophy we present here a new

group of transformations which allows us to find solutions in a simpler

way without reducing the number of independant variables. The invariance

of the system of equations is only required "in part" and source terms

can appear in the transformed system. Due to this fact groups of trans-

formations like these wi l l be called "GROUPS OF PARTIAL INVARIANCE".

The newness here consists in a rescaling of the time through a function

of t .

For a phase space system the following transformations

are defined

t = d t

C2(t)

[721 ~ : A(t)

: B(t) ~ + D(t)

189

This transformation is somewhat connected to transformation

introduced by Courant and Snyder (19) and rediscovered by Lewis (20)

Here we show how to use i t to solve some problems.

In order to keep invar iant a system of equations as much

as possible, the transformations defined by 172] are required to

ver i f y

1731 d ~ d ~ : d ~ d ~

that is to say, the element of volume in phase space must be formal ly

invar iant . This means that the Jacobian of the transformations must

be one. On the other hand, i f the system derives from a Hamiltonian

we impose two supplementary conditions

1741

:

=

where ~ and e are respect ively the new ve loc i ty and f i e l d . Notice that

we do not impose the s t r i c t formal invariance of Hamilton's equations.

The res t r i c t ion ]73[ and [74[ imply

A(t) D(t) : I • C2(t) dA B(t) • A(t) C2(t) : D(t)

then [72] becomes

do 1 - ~ :

C2(t)

175[ _ I

C ( t )

= C(t) ~ - dC

190

I t can be eas i ly proved~21jl ~ that th is set of transformation

obeys the law of group and generates a continuous Lie group, C(t)

being an a rb i t ra ry function.

These transformations can be generalized to the canonical

variables in analyt ical mechanics with a new parametrization of the

time (22) by wr i t ing

d9 1 =~

d t CZ(t)

1761 Qi - I qi C(t)

dC Pi = C(t) Pi - d-t qi

and i t can be immediately ve r i f i ed that they keep formally invar iant

the Poisson brackets

{~,~}pQ aH 3 F aH aF

~H @F BH ~F _ {H,F}pq ~Pr Bqr @qr @Pr

where H(Qr,Pr,8) : g H(qr,Pr, t ) is the transformed Hamiltonian

(g being any element of the group), and F is any function of the

independent var iables. The Hamiltonian remain s t r i c t l y invar iant only

for the t r i v i a l case C(t) = I ( i . e . for the neutral element of the

transformation group).

Nevertheless, we can define a new Hamiltonian ~{ with

(~}-[ # gH) so that a system of Hamilton's equations can be wr i t ten

]77L Qr = ~-T r

Pr °

191

I t contains new interact ions. Some can be traced back to the physical

in teract ions, others en t i re ly new, are connected to the change of

phase space. In th is sense the equations derived through the trans-

formations 1751 represent a new description of a physical problem

with a "renormalization" of the forces.

An interest ing fact arises from th is descript ion. I f the

function C(t) , characterizing the transformation group, increases

faster than t I /2 when t ÷ ~, the transformed time @ goes to a f i n i t e

value. In th is case we w i l l say that the time is renormalized in

the transformed system, which makes possible sometimes drast ic

s impl i f icat ions for numerical calculat ions.

There exists a very frequent l im i t i ng case obtained for

C(t) = ( I + Qt) 1/2 with ~ an arb i t ra ry constant. This par t icu lar

function generates a logarithmic compression of the i n i t i a l time.

For a form of C(t) with a leading term of degree smaller than one

ha l f the renormalization cannot be obtained and this type of trans-

formation group becomes usually ana ly t i ca l l y less in terest ing. (But

s t i l l in terest ing from a numerical point of view).

Another advantage of th is method is the poss ib i l i t y to

conter balance the time dependence of forces by a proper choice of

C(t) and to obtain time independent Hamiltonian or at least one

evolving in a slower way compare to i t s previous behaviour. We take

advantage of th is fact to solve th is "new" problem.

Let us f in i sh with a simple comment ; i f the time renorma-

l i za t ion is possible (without introducing i n f i n i t i e s on the forces)

we automatical ly can obtain informations on the temporal asymptotic

l i m i t for a l l i n i t i a l conditions, without e x p l i c i t l y solving the

equations. See paragraph below.

b) Example of a non-l inear harmonic osc i l l a to r

These groups of par t ia l invariance have useful applications

not only for the phase space f lu ids but for those systems in which

a renormalization of time or forces can be obtained. We are going

to comment here some interest ing results and develop two par t icu lar

192

examples ; namely the non- l i nea r harmonic o s c i l l a t o r and the non- l i near

heat d i f f us i on equat ion.

Let us s t a r t wi th the harmonic o s c i l l a t o r . Solut ions wi th

time dependent frequency have been obtained (23) fo r the l i n e a r cases

character ized by the equat ion.

d 2 X(t) + 2 ( t ) X(t) = 0 dt 2

wi th frequencies expressed by

2

1781 2 ( t ) = mo (1 + ~t) 4~ ; ~ > 0

mo and ~ are two constants.

Choosing f o r C(t) the fo l l ow ing form

1791 C(t) = (1 + ~t) B

= ~ fo r 0 < ~ < 1/2

wi th = 1/2 fo r 1/2 < ~ < I

= i f o r I <

We obta in a time compression fo r ~ < I / 2 , a logar i thmic compression

f o r 1/2 < ~ < I and a t ime renormal iza t ion fo r I < ~.

193

In a simi lar way i t is possible to study the non-linear

osc i l la tor defined by

1801 X + X + k X 3 = 0

where k is an arbi t rary constant, m(t) being a real function of time.

Taking for m(t) and C(t) the same expressions 1781 and 1791 the

following transformed equation is derived from 1801 by application

of the transformations 1751

1811 d2~ [B(~ 1) Q2 ( I + Qt) 4B - 2 Wo2(i Qt)4(B - m)] - - + - + +

dO 2

+ k mo2(1 + Qt) 6B - 4m~3 = 0

This equation can be interpreted as a non-linear osc i l la tor with a f i r s t l inear term ~2 ~ with ~2 given by

~ 2 ( t ) = mo2(1 + Qt) 4(B - m)

completed by a time dependent l inear force

- C 3 d2C ~ = -~(B - 1) ~2 (I + ~t) 4B - 2 dt2

which wi l l be called "transformation f ie ld" because i t is a v i r tual

f ie ld created by the transformation 1751, and f i na l l y the last term in the le f t hand side of 1811 which is the non-linear term. The value

of B must be chosen in order to obtain an equation without i n f i n i te

terms in the f ie lds and to renormalize the time to simpli fy the

194

numerical treatment. This las t requirement implies a value of

as high as possible within the l i m i t of the f i r s t requirement which

must be f u l f i l l e d f i r s t . With th is "strategy" table I shows the

d i f fe ren t choices of ~ when m runs from zero to i n f i n i t y .

TABLE I :

c~ 6

0 <c~ <3/4 2a

= 3/4 1/2

314<~<3/2 i12

= 3/2 1

> 3/2 1

C6(t)m2(t)

( l+~t) 3-4~

C4(t)m2(t)

4 -~-a (1+2t)

( l+2t) -1

2 - 4 ~ (1+2t)

( l+2t) -2

C3(t) C°( t )

8a-6 _ 2 ~ (2a-3)22(l+~t) ~

1 22

1 22

1 0

( l+~t) 6-4~ ( l+2t) 4(1-~) 0

2 0

i-~ (l+~t) 3 - 1

i - ~ - ~

log( l + at)

log( l + 2t)

2 t 28 = T + ~t

i f t ÷ ~

~ e + l

We note here some essential differences with the l inear

case (see (23)) beginning by the d i f fe ren t values of parameter

for which a renormalization of time is obtained.

We dist inguish the fol lowing cases :

A) 0 < a < 3/4. Due to the choice of 6, the term C6(t) m2(t)

becomes constant while the l inear force C 4 2 ~ is always smaller than

1 and decreases to zero with 0 ÷ ~ together with the transformation

f i e l d - C3~ ° ~. In the l inear case for 0 < a < I /2 the physical force

195

2 C 4 remains constant which indicates a solution x( t ) with i n f i n i t e

amplitude when t goes to i n f i n i t y . However in the non-l inear case the

term C 6 2 is dominant over the l inear one.

Consequently we wri te the equation which defines the l im i t i ng

t ra jector ies of the part icules when t and e ÷

1821 d2~ + k m 2 ~3 0 d~2 o =

d~ plane) can be and the asymptotic t ra jector ies ( l i m i t cycles in the ~

computed ana ly t i ca l l y .

B) 3/4 < m < 3/2. A renormalized time could have been obtained

i f we had taken 6 =~,~ Unfortunately the transformation f i e ld - C3C ° 1

would be divergent with the subsequent d i f f i c u l t y of computation.

Consequently we take the value 6 = 1/2 and obtain a logarithmic compression

of time.

Now the leading term is the transformation f i e l d and the

new asymptotic equation is simply

d2~ = 1 Q2

de--~ #

C) m > 3/2. I t is now possible to obtain a time renorma-

l i za t ion ( i . e . ~e ÷ I when t ÷ ~) by taking 6 = 1. The par t ic les go

to l i m i t points ~I in the ~-space. Their asymptotic behaviour is

d i rec t l y calculated from the transformation i t s e l f

Xlimi t = (1 + ~t) ~I

This aspect of the problem shows the advantages of the

method since now i t is only necessary to solve the equation

196

[83[ d2( : 0 + (1 - £e) 4(~ ' I ) ( + ( i - ~e) 4 ~-6 ~3

from e = 0 to @ = Q-1

to obtain asymptotic solut ions. In th is in terval the eq. 183] does not

present any i n f i n i t e term, and then there w i l l be no d i f f i c u l t y in the

numerical calculat ion.

c) Example of the non-l inear heat equation

Evidently, the main important feature presented by th is

type of transformations is the p o s s i b i l i t y of f inding asymptotic

solutions to a physical problem, even, i f the time renormalization

is not possible.

We shall return now to the non-l inear heat d i f fus ion

equation defined previously

8u k ~-~ (u s Bu ~-~ : V~)

The transformation groups of par t ia l invariance w i l l be applied to

obtain a transformed equation. Let us assume a transformation of

the form :

8 = 9(t)

: x . C ( t )

@(8,~) B(t) = u(x, t )

197

where B( t ) , C(t) and e( t ) are in p r i n c i p l e a r b i t r a r y funct ions, The

heat equation becomes

dB de 2¢ dC ~ 2¢ BS+l 2_~ 2¢ 1851 d ~ ¢ + B ~ - ~ + ~ ~ ~ = k C ( ¢ s )

o ~ o P a r t i c u l a r i z i n g : B Be and = #Ce, wi th C = B = 1 fo r t = e = 0

in order to obtain the same i n i t i a l condi t ions in the two spaces,

we get

B = exp 6 e ; c = exp y e

o

Then, a f t e r d i v i s i on by B O, the equation 1851 becomes

2¢ 2¢ exp (s6+2y)e @ 2¢ 1861 m ¢ + Y ~-~+~ = k ~ ~--~ (¢s T~ )

Choosing the funct ion 6 in such a way that

1871 de dt - exp (s# + 2y)e

the expression of t as a funct ion o f e is

1

s6 + 2y

A compression of time is obtained when we take

2y + s6 = -~ with ~ > 0

198

which gives the usual logarithmic compression

]88[ #0 = Ig(1 + pt)

Now we ask for conservation of the heat flow in th is

transformed space. Integrat ing the function u(x , t ) over the whole

x space, we obtain

/ u d x = / 4 d ( ~ = exp ( ~ - y ) 0 / ~ d~

I f we want the d i f fe ren t physical aspects to be the same

in the two spaces, @ must be taken equal to zero as ~ goes to i n f i n i t y

and imposing

Q = / u d x = / @ d ~ = ~

(Q being the quanti ty of heat), we must take

s + 2

~ 0, as i t can be easi ly seen by integrat ion in order to obta ins- C =

over the whole ~ space of equation 1861 with the condit ion 1871.

Then the non-l inear heat d i f fus ion equation is wr i t ten

- ~ (~ + ~ B~ Bq~ k ~ (q5 s ~q~ 1891 s + 2 T~ -) + ~ = ~ )

199

and the transformations of par t ia l invariance become

19oE ~ ~ = I g ( l + u t )

x : ~(I + ~t) I/s+2

u(x, t ) : #(~,0) ( I + ~t) -Vs+2

Let us study now the stat ionary solutions of eqo 1891 in

order to compare with the solutions obtained through the se l f s imi la r

groups. Imposing~-~ = 0 and integrat ing 1891, we obtain

s +-E ~ # = k#S aq~

As i t has been done in the se l f - s im i l a r case, we take only

those solutions with a maximum for x = ~ = 0 and the solutions obtained

are wr i t ten

_ Us 1/s : (K ~ ~2)

where K is a posi t ive constant and k has been taken equal to 1. We

note (as said previously) that the stat ionary solutions of 1891 are

ident ical to se l f - s im i l a r solutions obtained from 1211 when w is

taken equal to 1/s+2 (m being the arb i t ra ry parameter in se l f - s im i l a r

transformations), and i + ~t is replaced by t /T.

200

C O N C L U S I O N

We have presented in th is paper two kinds of philosophy

on the use of group theory in Mathematical Physics, in view of

get t ing solut ions for non- l inear par t ia l der ivat ive equations.

On the f i r s t philosophy we use group of transformations

leaving the equations s t r i c t l y invar ian t . Their use is l imi ted by

the fo l lowing reasons.

÷ They t reat res t r i c ted i n i t i a l or boundary condit ions and

i t is d i f f i c u l t to say i f s l i gh t change in these condit ions w i l l or

w i l l not modify considerably the structure of the solut ions

÷ I f we use Lie group the equations to be solved are as

d i f f i c u l t (sometimes more) than the i n i t i a l equations

Among the possible easy to sort out transformations we

must d is t ingu ish the s t retch ing ( se l f s imi lar ) ones. They are indeed

very easy to obtain and sometimes give se l f s im i la r solut ions of some

physical in te res t .

To balance these d i f f i c u l t i e s we notice

÷ That for some problems the d i f f i c u l t i e s of the se l f s im i la r

solut ions (usual ly divergence of some physical quant i ty at i n f i n i t y ) can

be taken care of through the concept of contamination where we remark

that the central part of a system with a s u f f i c i e n t degree of symmetry

w i l l not be immediately inf luenced by external parts (and sometimes

w i l l never be : th is is the important contamination concept).

÷ That se l f s im i la r group techniques are qui te in te res t ing

to search out the number of degree of freedom in the scal ing of a

system, and also to see, among many parameters, how many may be chosen

independently and what subsets are permissible.

In the second philosophy we drop the s t r i c t invariance

concept for the par t ia l invariance one. Usually we change a problem

in to a problem of s im i la r physics but where new terms make the system

201

easier to solve (and by easier we include also a numerical s imp l i f i ca t ion ) .

÷ The time compression or time renormalization is the most

important of the new concepts. Sometimes we can accelerate the time

without introducing i n f i n i t i e s in the forces or in the d i f fe ren t terms

of the equation. Then the usefulness is basica l ly numerical. In the

heat d i f fus ion equations ( l inear and non-linear) a time logarithmic

compression is quite obvious.

Of course, a time renormalization is s t i l l more interest ing

since i t gives d i rec t l y the asymptotic solut ions. In both cases

(compression or renormalization) the transformation is quite useful to

compute the asymptotic form.

The f i e l d of applications are quite numerous. Here we have

presented mostly applications to the evolution of phase space f lu ids

(plasma, beams, gravi tat ionnal gas where, inc ident ly , plasma are more

d i f f i c u l t to t reat than the two others phase space f l u i ds ) . Equations

in configuration space can also be transformed. Problems involving

di f fusion equations and Schroedinger equations have been solved by the

method (23). I t w i l l be interest ing to see i f hydrodynamics -which

already make good use of se l f s imi la r solut ions- is also an interest ing

f i e l d for the application of these new techniques.

Last -but not least- the inclusion of these transformations

within a numerical scheme is cer ta in ly an interest ing application

with i t s in t r igu ing merging between analyt ical and numerical methods.

202

B I B L I O G R A P H Y

10

11

G. Kalman ; Annals of Physics 1(], 1, (1960)

R. Balescu : "S ta t i s t i ca l mechanics of charged par t i c les" .

Wiley and Sons (London 1963)

D. Anderson, A. Bondeson, M. Lisak, R. Nakach and

H. Wilhelmsson : EUR-CEA. FC 925 (October 1977). To be

published in Physica Scripta.

F. Grant and M,R. Feix : Physics of Fluids 1(], 696 (1967)

H.L. Berk and K.V. Roberts : Physics of Fluids 1(], 1595 (1967)

S. Lie : "Theorie der transformations gruppen" I , I I , I I I .

Chelsea Publ. Company. (New-York 1970)

L.V. Ovsjannikov : "Group propert ies of d i f f e ren t i a l equations"

Translated by G. Bluman (1967), Cal. Inst. of Tech.

W.F. Ames : "Non-l inear par t ia l d i f f e ren t i a l equations in

engineering", Vol I I , Academic Press (1972)

H.S. Woodard : "S imi la r i ty solut ions for par t ia l d i f f e ren t ia l

equations generated by f i n i t e and in f in i tes imal group".

PHD Thesis Universi ty of lowa (1971)

G.W. Bluman and J.D. Cole : "S imi la r i ty methods for d i f f e ren t i a l

equations". Appl. Math. Sci. 13 Springer Verlag,

N.Y. Heidelberg Berl in

J.R. Burgan, J. Guti~rrez, E. Fi jalkow, M. Navet and M.R. Feix

Journal de Physique Lettres 38 161 (1977)

12

13

14

15

16

17

18

19

20

21

22

203

W.F. Ames : "Non-linear par t ia l d i f f e ren t i a l equations in

engineering". Vol I page 153 Academic Press (1965). The

solutions obtained in th is reference are found with pa r t i -

cular auxi lary conditions

N.N. Bogoliubov : "Studies in S ta t i s t i ca l Mechanics"

(Translated by E.K. Gora), Part A, Vol I . North Holland

Publishing Co. (Amsterdam 1962)

N. Rostoker and M.N. Rosenbluth : Physics of Fluids 3,

I (1960)

J.R. Burgan, J. Guti~rrez, E. Fijalkow, M. Navet and

M.R. Feix : J. Plasma Physics 199, 135 (1978)

A. Munier, M.R. Feix, E. Fijalkow, J.R. Burgan, J. Guti~rrez :

"Time dependent solutions for a se l f gravi tat ing star system".

Presented to publ icat ion.

J.W. Connor and J.B. Taylor : Nuclear Fusion, 1__77, 1047 (1977)

J.W. Connor and J.B. Taylor : Private communication

E.D. Courant and H.S. Snyder : Annals of Physics 3, 1 (1958)

H.R. Lewis and W,B. Riesenfeld : Journal of Mathematical

Physics 10, 1458 (1968)

J.R. Burgan, J. Guti~rrez, A. Munier, E, Fijalkow and

M.R. Feix : "Group transformations for phase space f l u ids " .

Note technique CRPE/47, page 9 (C.R.P.E./P.C.E. Orl~ans 1977)

J.R. Burgan : "Sur des groupes de transformation en physique

math~matique" P.H.D. Thesis. Universit~ d'Orl~ans (1978)

204

23 J.R. Burgan, M.R. Feix, E. Fijalkow, J. Guti~rrez, A. Munier :

"U t i l i sa t i on des groupes de transformation pour la r~solution

des ~quations aux d6riv~s pa r t i e l l es " . Proceedings de la

premiere rencontre i n te rd i sc ip l i na i re sur les probl6mes

inverses. Springer Verlag (1978).

24 G.L. Camm : Self grav i ta t ing star systems. Mont. Not.

Roy. Soc., 110, 306 (1949) ; 112, 155, (1951).

NONLINEAR KINETIC EQUATION IN PLASMA

PHYSICS LEADING TO SOLITON STRUCTURES

Carlos Montes Laboratoire de Physique de la Mati~re Condens~e, Parc Valrose, 06034 Nice

and

Observatoire de Nice, 06007 Nice

Induced Compton scatter ing of photons - and nonl inear Landau damping of plasmons -

by plasma par t ic les are governed by the same type of nonl inear i n teg rod i f fe ren t ia l

k ine t ic equation. Numerical time evolut ion of a large class of i n i t i a l spectra leads

to so l i ton l i ke behaviour. Only a f i n i t e number of exact invar iants of motion ex is t

and co l l i s i ons between sol i tons are performed in order to test t he i r s t ructura l sta-

b i l i t y .

I . INTRODUCTION

Up to now, only completely integrable models can be s t r i c t l y ana l y t i ca l l y invest igat -

ed. Only so l i t a r y waves sa t is fy ing completely integrable equations (which are solved,

e.g. , by the inverse scatter ing method) were cal led "so l i tons" . However, there is a

large class of nonintegrable physical systems studied by computer experiments which

lead to "so l i ton l i ke" behaviour. Following Makhankov 1 we may generalize the de f i -

n i t i on of a so l i ton as a so l i t a r y wave having some "safety factor" which ensures

a weak change in in terac t ion or c o l l i s i o n with others.

The purpose, here, is to show the time evolut ion of a large class of i n i t i a l boson

spectra governed by a nonl inear i n teg rod i f fe ren t ia l k ine t ic equation. The numerical

computation leads to so l i ton l i ke behaviour and approximate analy t ica l forms - in

the presence or in the absence of a constant noise spectrum - are exhib i ted. The

s t ructura l s t a b i l i t y is tested by so l i ton co l l i s i ons . Paired co l l i s i ons between boson

sol i tons can take place at a f i n i t e range in the same way that par t ic les having a

potent ia l . The existence of two exact invar iants of motion forces the two outgoing

sol i tons to be equivalent to the incoming ones a f ter exchanging t he i r i den t i t i e s .

Mul t ip le so l i ton co l l i s i ons , however, v io la te th is ; e.g. ternary co l l i s i ons show

the r ise of a much smaller fourth "so l i t on" .

206

I I . NONLINEAR KINETIC EQUATION

One d~mensional induced scattering of bosons (photons or plasmons) by a gas of non

re la t i v i s t i c particles is governed by a nonlinear integrodifferential kinetic equa- 2-6 tion of the form

F ~N(x,~.) . . . N(x,~,) oLx' WCx, x') N(x~,) (1) ~b

where N (X , t ) is the high level boson occupation number (the spectral maximum

sat isfy ing Nmax>> I) and where

~ I X , X') - " - - ~S/(X~ X) (2)

is the antisymmetric transit ion probabil i ty kernel. In what follows we shall take

WCx, x') = d G(x-x') (3)

where G(x- X') is a function tending rapidly to zero for [ x -X ' l - ->~ with a width

Z~ . Then, for X 2~> Z~ , the lower bound of the integral in (1) can be replaced

by-oo without appreciable error. This avoids the problem of boson condensation near

X = 0 . Equation (1) has two exact invariants of motion, namely

I~ : N (x, ~) dx (4)

L [2= 1o 9 N(×,'L)dx , (5)

and the following expression related to invariant (5)

dJ d j_o: :_ x 7 o ~t Nl ( × -t ) d ,e = - < G > I ~ (6)

where Z~ < G > = d ~ GC~) (7)

o :

I 1 and 12 seem to be the only exact invariantsof Eq. (1). Other exact invariants have

not been found.

For the case of induced Compton scattering of counterstreaming photons by Maxwellian

electrons, W(X,X J) is of the form 4,5

=-- e~p - ( --~-- 1 ~(x,x') ~A J× (8)

207

where X =~ IPo is the reduced frequency, Z~= 2(Vth/C ) is the Doppler width and

Wo= (ne OrT hc/meC2) ~o is the Thomson t rans i t ion probabi l i ty strength. For the case

of nonlinear Landau damping of plasmons by Maxwellian ions, the kernel (8) may be a good approximation 3,7,8 by taking × = k/kDe , as the reduced wave number (kDe being the Debye wave number), A2=m/M as the electron/ ion mass ra t io , and

standing for the normalization of the energy density per dk of the f luctuat ing elec-

t r i c f i e l d by the plasma k inet ic energy density (~pbeing the plasma frequency, n e

the electron density and T e , T i the electron and ion temperatures).

The parameters ~A/o and /~ in kernel (7) are useful for the physical in terpretat ion

of the scatter ing problem, however, Eq. ( I ) may be transformed into an universal form

without any parameter. Indeed, defining

: x / A (9)

: W o

and taking into account (8), we obtain

- (11)

I I I . BOSON SOLITONS

I t has been shown 4,5 , by a numerical treatment of Eq.(1) with kernel (8) , with

and without a source term, that in the presence of a constant noise spectrum ~Jo ,

whichever nonsingular i n i t i a l condition N I ~ , O ) transforms with time into so l i ta ry

waves moving downward on the frequency and/or wavenumber axis at constant speed 17"

and constant amplitude N m . A narrow highly populated (Nmax/N o >>I) i n i t i a l spec-

trum, centered at ~ =~o and having a width ~ < < I , transforms, with time, into

one boson sol i ton, i t s logarithmic spectral in tens i ty taking the approximate asymp- to t i c form 4,5

The speed 17" : I c~m/~11" I of the sol i ton moving downward on the ~ axis (where ~,~

is the abscissa of the maximum) may be found by putt ing c)/c)%- = o ~ ~//c)

into Eq. (11) and integrat ing twice with respect to ~: from -oo to ~ and from -co

208

to @~ , yielding

L i f o

(7" : -- -- (13)

f_~ 70 9 N(~$,~) <t~ f~

Using the approximate expression (12), the invariant expressions 11, 12 and O" are

given by

(N../NolJ -~ (14)

<,--- N°/[10 C N /No>]

Let us obtain the speed (16) in a different way which wil l be later useful in the

study of the time evolution in the absence of a constant noise spectrum.

For log (Nm/No)>> 1, we expand (12) as follows

(17) " N o e x p { 7 o ~ (N~/No)[t-(~-}o+~'~) 2]

The narrowness of the soliton profi le - i ts width is smaller by the factor

[log(Nm/No)]-l/2 than the Gaussian transition probability w id th - allow us to in-

troduce the l imit expression of (17), namely

NN~, N (~,~) --> ...... ~ ( ~ - ~o+ ~'~)

[30~ (N.,/No)}~,, ~ (1~) into the integrant of equation (11) and integrating over we obtain

By taking into account the soliton form (12), this may be written as an equation of

linear uniform motion in the ~ axis

Io9 FN ~,~)/No] = ~ --~ Io~ fNC~,~)/NoJ (20)

209

where the sol i ton speed ~ turns to be given by (16).

The nonlinear structure of Eq. (11) is responsible for the noise dependence of the

sol i ton motion. Indeed, i f the spectrum vanishes the motion stops. Therefore, there

is only uniform motion in the presence of an uniform noise spectrum No#O , which

uniformly ensures the nonlinear and nonlocal coupling of the sol i ton spectrum with

the noise spectrum.

Narrow spectrum evolution :

Figure 1 shows the numerical computation of the time evolution of an i n i t i a l narrow

Gaussian spectrum :

N / ~ . o l = N~ e~p [ - ( ~ - ~ o ) ' / ~ 1 + No (~1)

of width ~= 1/50, maximum amplitude Nm=108 , in the presence of a constant noise

spectrum No=l . The spectrum N ( ~ , T ) shows two main k inet ic regimes : i ) First~ the

I I ...... i , l , , , , , l , , l , , , , l l ' l ' J ' ,

- - 8 .

, - , . . '~%. f "%

- ." / \ / \ ., 6. • • ; ". /

" : ' I \ ." ": : / "" 4 ' ' g - . .: :1 ~: / ~ Z

- ." '. ~ \ t J - - ' : ' .. . . . / %/" '

• ' " . ' " , - - . . / ". ~ . / ,, - ~ ' ~ . : . . _ . . . . . . . O. Z

\ . I~! !i_ o m

m

_1

. . . . • " : 0 . 0 0

- - - . - "~': 2 . 9 5 . . . . . , t " = 2 3 . S l . . . . . . ,t.,,= 5 $ . 2 4 . . . . . . . . • r " : 9 6 .3~'

. I I I I I .1 I I .... 1 I ,I 1 I I i ] 8. 6. 4.. 2. I

i !~.2.

i

.os Oo -. os

~ o - ~

Fig.1 : Time evolution ofanarrow Gaussian boson spectrum of width 6 = 1/50 and amplitude N =i0 e on a noise spectrum N_=I . Logarithmic scale in ordinate ; two m d i f fe ren t real scales in o abscissa. The spectrum becomes with time ~= ~/~o a l e f t s a t e l l i t e and afterwards a sol i ton of amplitude Ns= 7.94xi06 moving downward on the ~ axis E~o = 2 r x v ~ ~1oeJ.

210

major part of the i n i t i a l spectrum transforms with time at a f i n i t e distance into a

s a t e l l i t e on the l e f t wing ; i i ) Then, the s a t e l l i t e moves downwards on the ~ axis.

I t is only in th is asymptotic regime that i t takes the sol i ton p ro f i l e which can be

very well approximated in logarithmic scale by a Gaussian, according to the approxi-

mate form (12).

I I I I I ! I I I I I I .... I I I I I I I I

- . . ~ ' = 0 . 0 0 -

i': ' r " : S.S7 - 2,,10 6 i;" - - . . . . ¢" = | 1 . t 4 -

, " . . . . . . . . . . . . 't" =t6.71 ! r ~ . . .

! l I I . . -

:i i i ~ I \ -

t.',/ ', / \ /'. 'I, ~\ / \ -~o' :1 I J l ' 1"o.I ~ ) n ",,x I I I ~ ' / I I I I I I I ~ . - -

2.0 1.5 t.0 O.S 0.0 -0.5 ~,o-~

Fig. 2 : Time evolution of a narrow Gaussian spectrum of width ~=1/4V2 and amplitude N =I0 $ in the absence of noise spectrum. Real scales. I t move~ with time ~ ' : ~ / ~ o downwards on the ~ axis with a decelerated motion, increasing l i near l y i t s ampl!tude (dashed line) and narrowing its width ~o =2 ~x~x1Os].

211

In the absence of a noise spectrum [No=0 in (21)] , the numerical computation of

the time evolution of (21) yields a decelerated motion. Figure 2 shows i t in real

scale. I t happens as i f the "sol i tary spectrum" would move on i ts proper le f t wing.

Its maximum amplitude turns to grow l inearly with ~ - ~ o , i ts width narrowing and

i ts speed decreasing. We can look for an approximate self-similar form of this sol i -

tary spectrum by generalizing the preceding method. We define the function g (%")

as the primitive of the time dependent speed ~ ( ~ )

I/ 9 f ~ ) = O" (n~') t iT ' (22)

and we introduce the following l im i t expression for the spectrum

:

~ ( T ) ] (23)

into the integrant of Eq. (11). Performing integration with respect to ~ and with

respect to ~ , we obtain

where A measures the norm, i .e. the constant of motion 11

Changing ~ / ~ by ~ -1~ /~ in (24), yields

(25)

(26)

where the dot in g means derivation with respect to "U . Imposing the conservation

of the norm (25) , we are able to obtain a closed dif ferential equation for g (~ )

which solution wi l l give the time dependence of the speed ~ (q ' ) and the se l f -

212

s imi la r expression for the so l i t a ry spectrum. This las t , from (26), reads

Expanding (27) in the v i c i n i t y of } - ~ o ='g (~d) and imposing the conservation of

the norm, we obtain

9 g 2 = 0 (28)

which for a Gaussian i n i t i a l spectrum y ie lds

I . _ _ ) = o (29)

A f i r s t order approximate solut ion, giving the pr incipal asymptotic dependence of g

on T , may be obtained by neglecting the las t term in (29). I t y ie lds

9 = (3A ' : ' ) ' I /~ (30)

This s imp l i f i ca t ion is a poster ior i j u s t i f i e d since the neglected term turns to be

~/~2 = _2(3A~.)-I/3 ~0 as ~ ' - ~ . By i t e ra t ion , we can obtain second order approx-

imations. Numerical computation ve r i f i es with good accuracy the der ivat ive of expres-

sion (30), namely

~(~i= ~ ( r ~ : ( A / s ) ' / 3 r - 2 / 3 (31)

Broad spectrum evolution :

Numerical computation of the time evolution of a broad spectrum of the form (21) with

>> i shows the s p l i t t i n g of the i n i t i a l p ro f i l e into secondary ones or sol i tons 4.

In the i r motion downward on the ~ axis, they are ordered by the i r amplitudes, or,

that amounts to the same thing, by the i r speeds, therefore increasing the distances

between them, When they are well separated, each sol i ton approaches the form (12).

IV. BOSON SOLITON COLLISIONS

In order to test the structural s t a b i l i t y of these boson sol i tons - i . e . of these

so l i t a r y spectra which move uniformly on a constant noise spectrum - interact ion bet-

ween them has been performed. We recal l here the pr incipal results of the computing experiments presented in reference 6 .

213

0

4 Z 0

2 o

0 M

48

z 4 0

~ 2 o

' - 0

't"~ 32

"u Z4

z 8 -16

z 4

" - ' 0 ~ , 8

20 lS t0 5 0

Fig. 3 : Phase diagram ( ~ , ~ ' } o f a binary co l l i s i on between two sol i tons of amplitudes NI = 4.4678 x 107 , N 2 = 2.2766 x 107 sa t is fy ing NI/N 2 = 1.9624.

They completely exchange the i r i den t i t y at a f i n i t e in teract ion range during the encounte~ Same time scale as in Figs. I and 2.

214

Binary co l l i s ions :

In paired co l l i s ions , the sol i tons may e i ther exchange the i r ident i t y (strength and

speed) at a f i n i t e in teract ion range, as par t ic les having a repulsive potent ia l , or

may merge by passing through each other when the i r amplitudes are very d i f fe ren t . In

a l l cases there is total transformation : the outgoing sol i tons seem to be equivalent,

wi th in the accuracy of the numerical computationjto the incoming ones. They only ex-

periment an appreciable sh i f t in the i r motion downward on the ~ axis. The stronger

so l i ton with amplitude N I exhib i ts a downward phase sh i f t ~ - ~ i , whereas the

smaller one~ with amplitude N 2 , exhib i ts an upward phase sh i f t ~ - ~ 2 , both

sh i f ts being related, from Eq. (6) and approximate expression (14), by

log - - ( 3 2 )

log Thus, the existence of the two invariants of motion I I and 12 forces the outgoing

sol i tons to be equivalent to the incoming ones , i f the number of sol i tons is conser- ! i

ved. Indeed, l e t (N1,N2) and (N I~ N2) be the amplitudes before and a f te r the interac-

t ion of two co l l id ing sol i tons sat is fy ing N I> N2>~-N o = 1 . Far enough before and

a f te r the co l l i s ion , we have from (14) and (15)

E ) , / : = >-

i : t ~:1

I

(33)

(34)

i

I f n = 2, as N1>>1 and N2>~11Eqs. (33) and (34) may only be sat is f ied i f N 1 = N 1 l

and N 2 = N 2 o r N 1 = N 2 and N 2 = N 1 . T h i s i s n u m e r i c a l l y v e r i f i e d , as we can s e e

on the phase diagram of Fig. 3.

Ternary co l l i s ions : i

I f n=3, the system of Eqs. (33) and (34) does not uniquely determine N i ( i= l ,2 ,3 )

by simple permutations. The existence of only two exact invariants of motion cannot

constrain the complete permutation of the respective sol i ton parameters (amplitude

and speed) in a ternary encounter, even i f the number of sol i tons is conserved. Nu-

merical computation of the time evolut ion of three d i f fe ren t sol i tons sat is fy ing Eq. (11) has been performed. By adjusting the impact parameters we obtain the ternary co l l i s ion represented on Fig. 4 .

215

Q

Z

~6 Z 4

0 ~ 2 0

' - 0

z 4 o

~2 o

, ' - 0

0

o

T '25

50

4O

30

I0

Fig. 4 :

ZO t5 10 ~ 0

Phase diagram of a te rnary c o l l i s i o n between three so l i t ons o f ampl i tudes N I = 4.4678 x 107 , N 2 = 2.2766 x 107 and N 3 = 7.4541xi06~ . There is no t o t a l exchange

(or permutat ion) in the encounter. Small dev ia t ions from the i n i t i a l ampl i tudes and v e l o c i t i e s are measurable and what is more impor tant a smal le r four th s o l i t o n r i ses , t he re fo re v i o l a t i n g the conservat ion of the number o f s o l i t o n s . Same t ime scale as in F igs. I , 2 and 3 .

216

We can observe a small deviation of the scattering trajector ies with respect to

the complete exchange of the incident ones and what is more important the generation

of a small fourth sol i tary spectrum. This spectrum is great enough to become with

time (too long for the numerical experiment) a sol i ton, therefore violat ing the

conservation of the number of such solitons in the interaction.

ACKNOWLEDGEMENTS

I express my gratitude to Professor A.F.Ra~adawho gives me the possib i l i ty to present

these results to the assembly. I thank Dr. J. Peyraud for his interest in the

problem.

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