Integration of Nonlinear Equations of Mathematical Physics by Method of Inverse Scattering Method I

8/4/2019 Integration of Nonlinear Equations of Mathematical Physics by Method of Inverse Scattering Method I

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A S C H E M E F O R I N T E G R A T I N G T H E N O N L I N E A RO F M A T H E M A T I C A L P H Y S I C S B Y T H E M E T H O DO F T H E I N V E R S E S C A T T E R I N G P R O B L E M . I

V. E . Zakh arov and A. B . Shab a t

E Q U A T I O N S

1 . G e n e r a l O u t l i n e o f t h e M e t h o dIn 1967 a grou p of Princ eton the oret ical physi cists (Garduer, Green, K ruskal, and Miura [1]) dis-

covere d a new method of mathem atical physics, the method of the inverse sca tte rin gpr obl em. The newmethod enabled its inventors to integrate an equation long known in the theory of nonlinear waves, namely,the Korte weg -de Vri es (KV) equation

ut + uu= ux=,, = O' (1 )In 1970 the aut hors of the pre sen t pape r, using ideas advanc ed in a paper by P. Lax [2], integ rated

(see [3, 4]), with the aid of the inve rse probl em method, the equationiu t -4- u== -4- lu l ~ u = 0 , (2)

also widely used in the physics of waves in nonlinear media. In the succeedin g ye ar s new examp les w erefound of nonlinear equations in tegra ble by the inve rse probl em method, over twenty of them having physicalmeaning (for a survey of inte grabl e equations, see [5]).

In the present paper we give a general scheme for applying the inverse scatteri ng problem method tointegrate nonlinear differential equations,and we also present an algorithm for f inding equations which canbe so integra ted. The original ver sio n of this sch eme was prese nted by one of us in [6].

We cons ider a linear integ ral oper ator F acting on vector-v alued functions = {1 .. .. g:N} of thevaria ble x( -~ < x < + ~)

P~ = ~ F (x, z) ~ (z) dz. (3)The vecto r and the N x N matrix -funct ion F depend additionally on the two pa ra me te rs t and y.quel we assume that

In the se-

sup ~ I F (x, z ) I dz < ~ fo r all xo > -- ~. (4)x~xo x o

We consider the problem of repres enting the operator F in the following "factorize d" form:t + P = ( t + k ) - 1 ( t + ~ _ ) ; ( 5 )

Here I~+ and K are Vo lt er ra op er at or s, whe re K+Cx, z) = 0 for z < x, and K_(x, z) = 0 for z > x. The op-er at or 1 + K+ is in ver tib le. Multiplying Eq. (5) by 1 + K+ and assu min g z > x, we can ver ify tha t the ke rnelK+ satisfies the Gel 'f and- Levi tan equation

L. D. Landau Institute of Theoretic al Ph ysics. Bashkir State University. Translated fr om Funkt-sion al'n yi Analiz i Ego l>rilozheniya, Vol. 8, No. 3, pp. 43-53, July-Sep tember , 1974. Original ar ticle sub -mitted March 14, 1974.

19 75 Plenum Publishing Corporation, 22 7 West 17th Street, N ew York, N. Y. 10011. No part of this publication m ay be reproduced,stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming,recording or otherwise, without written permission o f the publisher. A copy of this article is available from the publisher for $15,00.

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eo

~" (x, z) "4- K , (x, z) I K+ (x, s) F (s, z) as = 0.x

(6)

F u r t h e r , a s s u m i n g z < x , w e o b t a i n K _ ( x, z) :

K (x, z) = F (x , z ) ~ K+ (x , s) F (s , z ) ds .x

: I f E q . ( 6 ) i s s o l v a b l e ( a s u f f i c i e n t c o n d i t i o n f o r w h i c h , f o r e x a m p l e , i s t h e p o s s i b i l i t y o f r e p r e s e n t i n g t h ek e r n e l F in t h e f o r m F = F I + F 2, w h e r e F 1 i s a p o s i t i v e o p e r a t o r a n d fl F 21I < 1 ) , t h e n t h e k e r n e l s K + a n d K _a l s o s a t i s f y t h e c o n d i t i o n ( 4 ). W e d e f i n e o n t h e f u n c t i o n s ( x , t , y ) t h e o p e r a t o r :

O O ~ O= ~ - ~ z - + ~ % - + o., o ,, = z- 5 7- ..H e r e a a n d fl a r e c o n s t a n t s a n d l i s a c o n s t a n t m a t r i x . W e c o n s i d e r t h e c l a s s o f d i f f e r e n t i a l o p e r a t o r s

, e , ,M , c o n n e c t e d w i th M b y th e t r a n s f o r m a t i o n. ~ = ( t + K + ) .W ( i + g + ) - L (7 )

w i t h a V o l t e r r a [ s e e E q . (5 )] o p e r a t o r o f t h e t r a n s f o r m a t i o n 1 + K + . M u l t i p ly i n g E q . ( 7) on t h e l e f t b y ( 1 +K + ) , w e o b t a i n

. ~ ( t + R + ) - ( i + k + ) . ~ = o . ( 8 )T h e c o n d i t i o n o f e q u a t i n g to z e r o t h e d i f f e r e n t i a l p a r t s i n t h e o p e r a t o r r e l a t i o n (8) e n a b l e s u s t o c a l c u l a t eM :

0 0~ = a -~ - + ~ -~-~ + /~ ,, , n - - 1On On-~-1L , = t O 7 + ~ ' u ~ ( x ) ~ "f f i f f i 0T h e c o e f f i c i e n t s U k(X ) o f t h e o p e r a t o r L c a n b e f o u nd i n t h e f o r m o f a s e t o f r e c u r s i o n r e l a t i o n s

(9 )

uo (z) = [Z, ~0],T L"~'-~

d % + i 1 i d , ~ o ] + ~ d ~ ,u~ (x) = ~ l "-~-z~ T L~, 77zs j l T +t -~- 7~- [ , - ~ - +

. . . . . , , , , , , , , , , . . . . . , , , , , , , , . ,

(lO)

o o ) ' K ( x , z ) [ .... i o ( x ) = K ( x , x ) .e r e ~ (x) = a~ oz

Since the relation (7) is linea r in M and I~, we can take ~I to be the oper ator0 ~o = ~ , i n 0 "~ . ( 1 1 )

n

T h e n ~ . a aa - ~ - + ~ - ~ - + ~ , ' Z = ~ , L n . E q u a t i o n (8 ) then has the f o r mn

OK+ 0 K ~ a t*a T - t - ~ - - ~ - L K + ~ , ( - - i )"-* ~ g + l , = O . ( 1 2 )

F o r a g i v e n o p e r a t o r l~I o f t h e t y p e ( 1 1 ) , E q . (1 2) a n d th e c o n d i t i o n ( 10 ) d e f i n e t h e c l a s s o f V o l t e r r a o p e r a t o r s1 + K +, t r a n s f o r m i n g M i n t o i ~.

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T h e f o ll o w i n g t w o t h e o r e m s p l a y a f u n d a m e n t a l r o l e i n t h e s e q u e l .T H E O R E M 1 . I f t h e o p e r a t o r F ; w r i t t e n i n ; th e f o r m ( 5L s a t i s f i e s t h e c o n d i t i o n

t h e n t h e o p e r a t o r s (1 + ~C ~) ~ w h i c h a c c o m p l i s h t h e f a c t o r i z a t i o n o f F , s a t i s f y t h e r e l a t i o n sM ( i + R e ) - - ( t + / ~ + ) ~ = 0

(13)

(14)and are operators transforming l~I into ~ .

To prove this we write the identity

~( t + /~ ) - - ( l+ k ) ~ r = ~ ( i + k + ) ( i + F ) - ( t + k + )( i + ~) ~ / - -= { ~ (i + R+) - (i + k+) ~ ) (i + g) + (i + g+) {~ P - PM}. (15)

B y v i r t u e o f t h e r e l a t i o n s ( 1 0 ) , t h e d i f f e r e n t i a l o p e r a t o r o n t h e r i g h t s i d e a n d , h e n e e~ a l s o o n th e l e f t s i d e o ft h e i d e n t i t y ( 1 5) is a b s e n t . F o r z > x th e l e f t s i d e i n t h e r e l a t i o n (1 5) v a n i s h e s , a n d f r o m t h e i n v e r t i b i l i t y o fthe ope ra tor 1 + F i t fo l low s tha t ~ ( i + k+) - ( i + R+) ~ f = 0 . Fu r th e r , i t i s obvious tha t ~ ( i h : ) - -(1 + K_) M = 0.

T H E O R E M 2 . S u p p o s e t h a t t h e o p e r a t o r F s a t i s f i e s s i m u l t a n e o u s l y t h e t w o r e l a t i o n s[~ t , / ? ] = 0 , M,=a~-~i--l-L(ol), (16)[M~,gl = 0,

H e r e ~ 1 ) a n d L ~2 ) a x e o p e r a t o r s o f t h e f o r m

0 _L P.(~),M2 ---- ~ ~- 7- ~0 (17 )

Nt NIL( :'= F l~' o" L~"= ~ z~' 0"s a t i s f y i n g t h e c o n d i t i o n

[/2 ), Lo(~)]= 0,T h e n t h e o p e r a t o r s L (1) a n d L (2) s a t i s f y t h e r e l a t i o n

(18)

I (19)

~-~-v0L(I) - a--6T-/'(~) [L (~), ?.(%P r o o f . B y v i r t u e o f T h e o r e m 1 w e h a v e t h e fo l Io w i n g r e l a t i o n s f r o m E q s . ( 17 ):

(20)

o + L(', A;/~ = Bo~ + L(') . (21)Writing out the relations (21) in explicit form, applying to the first of them the operator B(3/0y) and to thesecond the operator od0/0t), and then subtracting the second from the first, we obtain

(8 L


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a n d ( 17 ) c a n b e s o l v e d b y t h e F o u r i e r m e t h o d ; E q . ( 19 ) g u a r a n t e e s t h e i r c o m p a t i b i l i t y . T h u s , t o e a c h p a i ro f o p e r a t o r s L ~l ), ~.~2) t h e r e c o r r e s p o n d s a n t n t e g r a b l e s y s t e m o f t h e t y p e ( 2 0) . P u t t i n g a = 0 o r f~ = 0 , w eo b t a i n t h e e q u a t i o n s

-~-(1) _ iL(~), L(,)], (22)

a --gi--(~) ---- [L(~) ' ~(1)], (23)w h i c h a r e a l s o n a t u r a l l y r e l a t e d t o t h e g i v e n p a i r o f o p e r a t o r s L ~ ) , (2)L 0 2 . S c a l a r O p e r a t o r s

A s a f i r s t e x a m p l e o f a n a p p l i c a t i o n o f t h e p r o p o s e d s c h e m e w e c o n s i d e r t h e c a s e o f s c a l a r o p e r a t o r sL . U s i n g s u c h o p e r a t o r s , w e c a n i n t e g r a t e t h e e q u a t i o n s w h i c h a r i s e i n p h y s i c s i n c o n n e c t i o n w i t h p r o b -l e m s c o n c e r n i n g t h e p r o p a g a t i o n o f n o n l i n e a r w a v e s i n m e d i a w it h a w e a k d i s p e r s i o n . I n t h e s c a l a r c a s eE q s . ( 10 ) h a v e t h e f o r m

n da~o n d dEoU o ( X) = 0, u , ( x ) = n , u , 2 ( x ) = ~ ( n - - 2 ) ~ + , y - ~ - ~ l + n ~ o - ~ - x .F r o m E q s . ( 24 ) i t f o l l o w s t h a t L 1 = L 01. A s t h e f i r s t n o n t r i v i a l o p e r a t o r s w e h a v e

(24)

Lo2 = -~-z~ L~ = ~ + u (x), u = 2 ~o (x) = 2 K (x, z) ,o. a a~ 3( 0 Ou ) ~ oLo3 =-g~-~ + ~ - ~ , L3=-g;r~ +-X- u- ~- + 0x + w + 0x '

3 dw = -~- ~ (~1 + ~) . (25)AT h e c o m m u t a t o r o f t h e o p e r a t o r s L 2, L s h a s t h e f o r m

[?.2, L3l_= O.~_w w o _ t ~,u~].ox a~ [' T (u~'~ + 6uu~) +AS IVI1, IVl2 w e c o n s i d e r t h e f o l l ow i n g o p e r a t o r s :

(26)

0,~1 = a w + Lo3, ~ 2 = Lo2. (27)

H e r e t h e k e r n e l F s a t i s f i e s t h e e q u a t i o n s02F O~F8x~ Oz~ ~ O,

T h e k e r n e l K s a t i s f i e s t h e e q u a t i o n

OF . O'F . O~F ._ (O F OF) (28)

O~K + u (x ) K = O,Oz 2 Oz"f r o m w h i c h t h e r e f o l l o w s a c o n d it i o n o n t h e c h a r a c t e r i s t i c z = x

dd- ; (h + ~) = 0,

(29)

(30)

i nd ica t ing tha t w = 0 . E q ua t ion (23) now ha s the f o r m (~(1) = L s , L(2) = L2)Ou ta .~f + T (u~.~ + 6uu~ + ~,u,,) = 0 (31)

a n d c o n s t i t u t e s a n u n k n o w n K o r t e w e g - d e V r i e s e q u a t i o n . I t s h o u l d b e r e m a r k e d t h a t i n t h i s e x a m p l e b o t ht h e f u n c t i o n u ( x , t ) a s w e l l a s t h e c o n s t a n t s c~ a n d X c a n b e c o m p l e x . T h i s w h o l e d i s c u s s i o n c a n a l s o b e

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c a r r i e d t h r o u gh a s s u m i n g u ( x, t) t o b e a m a t r i x b e l o n gi n g t o a n a r b i t r a r y a s s o c i a t i v e m a t r i x a l g e b r a .M o r e o v e r , E q . ( 31 ) c a n b e w r i t t e n i n t he f o r mOu ta "g + "U (u= ~ + 3UU~ + 3Ug t + ~U=) = O.

o p e r a t o r s l V l1 and 1~I2 in the fo rme now c o n s i d e r t h e

J l----~03~T h e k e r n e l F n o w o b e y s t h e e q u a t i o n s

~'~ ---- ~~ -4- Lo,. (32 )

and Eq . (22 ) has the fo rm

OaF . OaF I OF O F \ o, ~ 0,1: O2F = O, (33)aF ~_ Oz~ Oz~

3 f~Ou Ow Ow t Uo= ' ~ ~ = "W ( ='~ + 6uu=) + ~,u= (34)o r

3 ~ o,,, i 3 (uu=)= = o.T ~ Ty, + ~,u= + T u=== + " TP u t t i n g 3 / 4 B2 = + 1 , X = * 1 , we ob ta in fou r r ea l equa t ions

O"-u O'u t 3-F ~ -4- -g-g + "X- u=~==+ 7 (uu=)= - O. (35)T w o o f t h e m ,

( O~u O2u ) t 3 (uu= )= = O,+- oy, o=, + T u=== +r e p r e s e n t v e r s i o n s o f t h e e q u a t i o n f o r a n o n l i ne a r j e t ( s e e [7 ]) . F i n a l ly , c o n s i d e r i n g t h e g e n e r a l c a s e a n dpu t t ing

0 8M1 = a ~/- + Lo3, M.. = ~ Ty + Lo~,w e o b t a i n f r o m t h e r e l a t i o n (2 0) t h e s y s t e m o f e q u a t i o n s

3 g 8u Ow Ow Ou i--~-,.-~ -ffi ---~-= , [~-~f ffi a- -~ + ~u= + -T (u = = + 6uu=), (36)e q u i v a l e n t t o t h e e q u a t i o n

3 - ~ u o c Ou t - t - 6uu= ) } = O.- - ~ ~- +T = (a -~ -+ Xu~ +-Z- (u=,= (37)I n a d d i t i o n t h e k e r n e l F s a t i s f i e s t h e t w o e q u a t i o n s

ap a,p a,r _X (O F o~ 1 OF a' r a , F = o "a " g / - + ' ~ - - + " g ' g ' + " ~'- = FT z / = 0 , P -~ -y --Fax , az, (38)E q u a t i o n ( 37) w a s o b t a i n e d f o r t h e f i r s t t i m e i n [ 8] ; w e s h a l l c a l l i t t h e K a d o m t s e v - P e t v i a s h v i l i e q u a t i o n .T h i s e q u a ti o n d e s c r i b e s a t w o - d i m e n s i o n a l n o n s t a t i o n a r y p r o b le m c o n c e r n i n g w a v e s i n a d i s p e r s i v e m e -d i u m , p r o v id i n g t h a t t h e s c a l e c r o s s w i s e t o t h e p r o p a g a t i o n o f t h e w a v e ( a l o n g t h e y a x i s ) i s m u c h l a r g e rt h a n t h e l o n g i tu d i n a l s c a l e ( a l on g t h e x a x is ) .

F r o m w h a t h a s b e e n s ai d i t is c l e a r t h a t e a c h p a i r c~ s c a l a r o p e r a t o r s L ~0, L~2) g e n e r a t e s i n e a c hm a t r i x a l g e b r a a n e q u a ti o n o f t y p e ( 20 ).

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3 . N - S o l i t o n S o l u t i o n sI n t h i s s e c t i o n w e d e s c r i b e a n i m p o r t a n t c l a s s o f p a r t i c u l a r s o l u t io n s o f t he K a d o m t s e v - P e t v i a s h v i l i

e q u a t i o n ( 37 ), t he N - s o l i t o n s o l u t i o n s . W e r e s t r i c t o u r c o n s i d e r a t i o n t o t h e s p e c i a l c a s e ~ = f l = 1 0 , < 0 ) .E q u a t i o n ( 3 7) t h e n d e s c r i b e s w a v e s o n t h e s u r f a c e o f a s h a l l o w li q u id ; i t is w r i t t e n i n a f r a m e o f r e f e r e n c em o v i n g w it h a s p e e d e x c e e d i n g t h e s p e e d o f i n f i n i t e l y lo n g w a v e s . W e p u t

F = e-~=-'aM (t, y). (39)Fr om Eqs . (38) i t fo l l ows tha t

M = M ( t , y) = M o e x p { ( q ' - - x 2) y + ( x 8 + ~I + X ( x + n ) ) t } . (40)S o l v i n g t h e G e l ' f a n d - L e v i t a n e q u a t i o n ( 6 ) , w e f i n d

M~-Xx-~*JzK (x, z) = M '1 + ~ e -~+ ~)x

u (x) = 2 K (x, x) = -~- ch~(x + n) (x -- xo) 't , MoXo = (~1 -- ) Y + ('q~ " x'q + ~ + ;~) t + ~ m "7 -~ " (41)

Theh e e x p r e s s i o n s (41 ) d e s c r i b e a s o li t o n , i . e . , a s o l i t a r y w a v e p r o p a g a t i n g a t a n a n g l e t o t h e x a x i s .s o l i t on i s c h a r a c t e r i z e d b y t h r e e p a r a m e t e r s : u a n d 7? c h a r a c t e r i z i n g i t s a m p l i t u d e , i t s a n g l e o f i n c li n a t io nt o t h e x a x i s , a n d i ts v e l o c i t y ; t h e p a r a m e t e r M o c h a r a c t e r i z e s t h e p o s i t i o n o f t h e c e n t e r o f t h e s o l i t o n w h e nt = 0 and y = 0 . When ~ = ~ , t he so l i t on p rop aga t e s s t r i c t l y a long the x ax i s ; when V2_uW + ~2+ ~ = 0 (~< 0 ) ,t h e s o l it o n i s s t a t i o n a r y i n t he g i v e n f r a m e o f r e f e r e n c e .

A s s u m e n ow t h a tF = ~a M. e-"'~-'~n'' (42)

n

As before, we haveMn (t, y) = M . (0) ex p {(1]~ - - ~) y + (z~ + ~1~ + ~ (Tin -k- .)) t}.

Pu t t i ng , a s we d id ea r l i e r , K (x, z ) = ~ K. (x )e -~ '~ , we o b t a in a sy s t em of equa t io ns fo r t he Kn(X) :n

-(Xn +V,m)xKn (x) + M~e -x,'x + Mn ~_~ e Km (x). (43).~ n + XlmT h e g e n e r a l s o l u t i o n o f t h i s s y s t e m h a s t h e f o r m

d -r,nX d ~u(x)-~2-d-~x~K.(x)e = 2-~rx~ n A,

A = detl3 LS. .~ ~-, M.e-t~n-~'~)~ll--.%, . (44)

E q u a t i o n s (4 4) g i v e a n e x p l i c i t f o r m o f t h e s o lu t i o n , w h ic h f o r t ~ + % y ~ ~ b r e a k s u p a s y m p t o t i -c a l ly i n to a n a g g r e g a t e o f n o n i n t e r a c t i n g s o l i t o n s p r o p a g a t i n g a t a r b i t r a r y a n g l e s t o th e x a x i s . A n a l o go u se x p r e s s i o n s f o r t h e K V e q u a t i o n w e r e f o u n d i n [9 ] . A n a t u r a l g e n e r a l i z a t i o n o f t h e s o l u t i o n ( 41 ) i s a s o l u -t i o n f o r w h i c h

F = (~ (x, y, t) e -*'z. (45)

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H e r e t h e f u n c t i o n ~v s a t i s f i e s t h e e q u a t i o n sa + _ ~ _ ~ aa--/" a ~ - - ' " a ~ - - (~ a + ~ ) ~ 0 = O , ~+ ~~ - , - - ~ = 0 ( 4 6 . )

a n d h a s t h e f o r m.Jo~

= I c ( k ) e x p { i k y - - x x - t - ( ~ + v l~ - 4 - X ( u ~ - ~ l ) ) t } d k ,

c ( - - k ) = c '( k ), x ~ = n ~ - - ~ k . ( 4 7 )H e r e

K ( x , z ) = q ~ ( z ) e ~ ' z u ( x , y ) = 2 d ~ , ( ~ ) e ' =d z' -~ I ~ ($) e-~$ d$ ' ~- I ~ ($) t~"tlSd$ ( 4 8 )

T h e e x p r e s s i o n s (4 8) d e s c r i b e a n " o s c i l l a t in g " s o l i t o n ; w e ca r t i n a n a n a l o g o u s w a y c o n s t r u c t g e n e r a l i z a -t i o n s o f ~ - ~ so l it o n s o l u t i o n s .4 . M a t r i x E q u a t i o n s

A s s u m e n o w t h a t L ~ ) a n d L I 2) a r e N N m a t r i x o p e r a t o r s o f t h e f i r s t o r d e rL ( o ~ ) = l l ~ . ~ , = ~ - , [ 1 ~ , / ~ 1 = 0 .

T h e n

= l l -g b- -4- [11, ~ol, 12-~z -4- [l~, ~o1. (49 )C h o o s i n g t h e m a t r i c e s ( 4 9 ) t o b e d i a g o n a l m a t r i c e s : l~ = b =6 ~r n, I s = a n f n m (a~ ~ = 0 ) , w e o b t a i n f o r E q . ( 2 1)( s e e [ 1 0 ] )

( a i - - a j ) ~ S i1 - -- - ( b i - b j ) O S i J ~ - ~ ' i ~ + ~ ( s i k - ~ s ~ j - 4 - s j i ) ~ i ~ j ~i1 _-- ~ . (50)

T h e s y s t e m ( 50 ), b e g i n n i n g w i t h N = 3 , i s n o n t r i v i a l . W h e n N = 3 , o f p h y s i c a l i n t e r e s t a r e t h e " r e d u c t i o n s "o f t h e s y s t e m ( 50 ) t o m a t r i c e s c o n t a i n i n g h a l f t h e n u m b e r o f i n d e p e n d e n t f u n c t i o n s , t h e s e b e i n g d e r i v a b l ew i t h t h e h e l p o f t h e r e l a t i o n s ~ + = I~ I , 12 = 1 , w h e r e I i s a d i a g o n a l ~ m a t r i x . I f I = 1 , t h e n t h e s y s t e m ( 5 0 ),b y m e a n s o f t h e s u b s t i t u t i o n ~ i j = ; ti ju i j (X ij a r e c o n s t a n t s ) , i s r e d u c e d t o t h e f o r m

O U l2a t - 4 - V I ~ V u l ~ = i u l s u * ~ , - - g ' F ' O u = 4 " V~3Vu2s ---- u 1 3 u l ~ , ~- Vi3Vu13 = i u l ~ u ~ a . ( 5 1 )H e r e th e V i j a r e c o n s t a n t t w o - d i m e n s i o n a l v e c t o r s . T h e s y s t e m (5 1) d e s c r i b e s t h e r e s o n a n c e i n t e r a c t i o no f t h r e e w a v e s i n a n o n l in e a r m e d i u m a n d p la y s a n i m p o r t a n t r o l e i n n o n l i n e a r o p t i c s . C h o o s i n g t he m a t r i x

I -- -- 0 - - t , w e c o m e t o t h e s y s t e m0 0

OUl~a t + V12V ul-. = ~ u ~ 3 u ~ 3, o , , ~ 0 ~ , 1 , _ V V u - -~ - + V~3Vu~ 3= ~ u 1 3 u 1 2 , " - ' g V - r 1 3 ~ = ~ u ~ t ~ , ( 5 2 )d e s c r i b i n g " e x p l o s i v e i n s t a b i l i ty " o f a n o n l i n e a r m e d i a m ( s ee [1 01 ). T h e r e m a i n i n g w a y s o f c h o o s i n g t h em a t r i x I l e a d t o s y s t e m s e q u i v a l e n t to th e s y s t e m s (5 1) o r ( 5 2 ).

I n t h e c a s e N > 3 , m o r e i n v o l v e d r e d u c t i o n s o f t h e s y s t e m (5 0) a r e p o s s i b l e ; t h e i r c l a s s i f i c a t io n w o u l db e o f g r e a t i n t e r e s t f o r t h e a p p l i c a t i o n s .

2 3 2


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T h e c a s e i n w h i c h s o m e o f t h e a i = b i = 0 i s a d e g e n e r a t e c a s e a n d i s a l s o o f g r e a t i n t e r e s t .c a s e th e o p e r a t o r s ~ ( 0 ~ (2 ) c a n b e r e p r e s e n t e d i n t h e f o r m

L , , [ L oLz 0 '

3 ~ 0 '

I n t h i s

" ~ o 0 ,= l x ~ ; -w [ /x , ~ n l ,

H e r e l l a n d / i a r e d i a g o n a l M x M m a t r i c e s n o t c o n t a in i n g z e r o s , a n d t h e ~ . . a r e c e l l s of t h e m a t r i x ~ . C o n -: . . . . 1 Js i d e r i n g o n l y t h e c a s e i n w h i c h t h e r e i s n o d e p e n d e n c e o n y , w e r e a d i l y s e e t h a t n o w t h e s y s t e m ( 50 ) r e -d u c e s t o t h e f o r m

0 h [ L o , 3 0 1 + t & q ] ,0~a - - - ~ = [ G , , 4 o l + L o q G - G q L o ;

= L~L~,q = lxl~x. ( 5 3 )

1 0 ]I n t h e c a s e N = 4 , M = 2 , I~ = 0 . -- 1 ' l : = i t h e s y s t e m ( 53 ) l e a d s t o t h e k n o w n " s i n - g o r d o n " e q u a t i o n( s e e [ 1 2 ] )

82u O*uOt'~ Oz~ q - s i n u - -- - O .

I t s h o u l d b e n o t e d t h a t in t h e d e g e n e r a t e c a s e (5 3) t h e p r o c e d u r e w e h a v e p r e s e n t e d i s o n l y s u i t a b l e f o rf i n d i n g i n t e g r a b l e e q u a t i o n s , s i n c e t h e f o r m a l a p p l i c a t i o n o f t h e G e l ' f a n d - L e v i t a n e q u a t i o n (6) i n t h e d e g e n -e r a t e e a s e l e a d s to d i v e r g e n t e x p r e s s i o n s . F o r th e " s i n - g o r d o n " e q u a t i o n t h i s d i f f i c u lt y w a s s u r m o u n t e d i n[ 1 2 ] .

M a t r i x o p e r a t o r s o f r ~ uc h h i g h e r o r d e r s h a v e s o f a r n o t b e e n s t u d ie d . A n e x c e p t i o n h e r e i s th e e a s ei n w h i c h L 2) = l ( 0 / 0 x ) , w h e r e l i s a m a t r i x o f t h e f o r m l = 0 12 'a r b i t r a r y s c a l a r o p e r a t o r . T h i s e a s e c o i n c i d e s in i t .s e s s e n t i a l f e a t u r e s w i th t h e p u r e l y s c a l a r c a s e . C a l -c u l a t i n g ~ ( 0 a n d ~ ( 2 ), w e h a v e

~a(1) 0 2 ,= -~-x2 ~ 2 ~, l l 0L (2 )= [ 0 l ~ ] + ( l l - - l " ) [ ~ O 1 ~ ! ] ,

a n d w e o b t a in f r o m t h e r e l a t i o n s ( 21 ) f o r f l = 0 a s y s t e m o f e q u a t i o n s f o r t h e a n t i d i a g o n a l c e l l s o f t h em a t r i x

o _~ t , + t.. o~ ! ~ : ~ _ 2 ~ g~xg:2~.,: O; (5 4)( 54 ) a d m i t s t h e n a t u r a l r e d u c t i o n ~2~ = ~ a n d r e d u c e s t o t h e s i n g l eo r ~ = i a n d r e a l l l , l 2 t h e s y s t e m

e q u a t i o n( 55 )

I n t h e s c a l a r c a s e E q . ( 55 ) c o i n c i d e s w i t h E q . ( 2 ). T h e m a t r i x c a s e o f E q . ( 5 5) w a s c o n s i d e r e d b y S. V .M a n a k o v i n [ 1 3 ] . I n t h e c a s e i n w h i c h L I l) = 0 3 / 0 x a t h e E q . ( 23 ) l e a d s t o th e " m o d i f i e d " K o r t e w e g - d e V r i e se q u a t i o n ( s e e [1 4 ] )

u , I u l 2 u ~ + u ~ = 0 . ( 5 6 )

C a s e s i n w h i c h t h e o p e r a t o r L I l) is o f a n o r d e r h i g h e r t h a n th e t h i r d a r e n o t o f i n t e r e s t f r o m t h e p o i n t o f v i e wo f t h e a p p l i c a t i o n s .

2 3 3


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5 . S o l u t i o n S c h e m e f o r t h e C a u c h y P r o b l e mW e n o w p o s e t h e q u e s t i o n a s t o th e p o s s i b i l i t y o f s o l v i n g th e C a u c h y p r o b l e m f o r t h e E q s . (20 ) an d

(2 3) w i t h in i t i a l c o n d i t i o n s i n t. L e t ~ 2 ) b e t h e o p e r a t o r d e f i n e d i n E q s . (1 81 , h a v i n g o r d e r N 2. T h e c o r -r e s p o n d i n g o p e r a t o r ]~ (21 c o n t a i n s N 2 - 1 v a r i a b l e c o e f f i c i e n t s u i (x , y , t ), w h i c h w e a s s i g n a t t = 0 . W e s h a l la s s u m e h e r e t h a t

ui ( d : c o , y , 0 ) = 0 . ( 5 7 )F r o m E q . (1 2) i t fo l l o w s t h a t t h e k e r n e l K + o b e y s t h e e q u a t i o n

N z~ 2~ 0 K + L ( ' ) K + - t - t ) ' ~ - x ' = K + l = 0 . 158toy + ~" ( - o=" '~* J l

O n i t s s o l u t i o n w e i m p o s e t h e r e s t r i c t i o nK + ( z , x +s)-~O for x --~ + co. (59)

W i t h r e s p e c t t o t h e f u nc t i o ns u i ( x , y , 0 ) w e c a n , u s i n g t h e E q s . ( I 01 , f i n d t h e d e r i v a t i v e s o f t h e k e r n e l K ~a l o n g t h e n o r m a l t o t he l i ne x = z . ~ 1 (x , y , 0 ) {i = 0 . . . . , N 2 - 2 ) , w h o s e a s s i g n m e n t , a l o n g w i t h t h e c o n d i t i o n( 59 ), d e f i n e s a C a u c h y - G o u r s a t p r o b l e m f o r E q . ( 5 8 ). T h e s o l u ti o n o f t h i s p r o b l e m d e t e r m i n e s t h e k e r n e lK + { x , z ) f o r z > x . I n p r o c e e d i n g f u r t h e r w e f o l l o w t h e s c h e m e

ui (x, y, 0 ) ~ K -+ (x, z, y, 0) ~ F (x, z, y, 0) I,I F (x, z, y, t) iv___.K + (x, z, y, t) v_._.u~ (x, y , t). (60)I n t h e f i r s t s t a g e o f t h i s s c h e m e w e s o l v e t h e C a u c h y - G o u r s a t p r o b l e m f o r E q . ( 58 ); in th e s e c o n d

s t a g e , w i t h th e h e l p o f t h e G e l ' f a n d - L e v i t a n e q u a t i o n ( 6) , w e d e t e r m i n e t h e k e r n e l F a t t = 0 . T h e e v o l u t io no f t h e k e r n e l F i n t i m e i s d e s c r i b e d b y t h e l i n e a r e q u a t i o n (1 61 w i t h c o n s t a n t c o e f f i c i e n t s , c o n s t i t u t in g t h et h i r d s t a g e o f t h e s c h e m e . I n t h e f o u r t h s t a g e w e u s e E q . ( 6) t o o b t a i n th e k e r n e l K + ( x, z , y , t ), a n d f i n a l l y ,a t th e l a s t s t a g e w e a p p l y E q s . ( 1 0) , e n a b l i n g u s t o f in d u ( x, y , t ) .

A f u n d a m e n t a l d i f f i c u l t y i n a p p l y i n g t h e s c h e m e (6 0) i s t h a t t h e k e r n e l K + ( x, z , y , t ) o b t a i n e d f r o mt h e s o l u t i o n o f t h e C a u c h y - G o u r s a t p r o b l e m (5 81 , (5 9) c a n n o t s a t i s f y t h e c o n d i t i o n ( 4) o f d e c r e a s e w i t h r e -s p e c t t o z . 0 ]

W e c o n s i d e r t h i s s i t u a t i o n b y a n e x a m p l e i n w h i c h / ~ ' ) = l o ~ , l = 12 0 , I I > 12 > l s > 0 , a n d B = o .I n t h i s c a s e E q . ( 58 ) h a s t h e f o r m 0 ls

K S x 5 ~ z (611

i j ( x ) e i ~ z , w e o b t a i n a s e t o f t h r e e s p e c t r a l p r o b l e m s f o r i j , w h i c h w e n u m b e r b y m e a n s o fs i n g Kijt h e s u b s c r i p t j ,

k(62)

w i t h t h e c o n d i t i o n s{plje~- -* 0 for x ~ -- c~,~ j l e 7 "= - - ~ 0 f o r x ~ ~- ~o. (63)

T h e a s y m p t o t i c b e h a v i o r o f t h e k e r n e l K i j f o r z ~ + oo i s d e t e r m i n e d b y t he d i s c r e t e c h a r a c t e r i s t i c n u m b e r sc f t h e s p e c t r a l p r o b l e m s ( 62 ), ( 6 3 ), l y i n g in t h e c o m p l e x p l a n e . F o r j = 1, 3 i t f o l l o w s f r o m t h e c o n d i t i o n s(6 3) t h a t I m ), > 0 f o r d i s c r e t e v a l u e s o f ~ , a n d th e a s y m p t o t i c b e h a v i o r i s d e c r e a s i n g . H o w e v e r , w h e n j = 2t h e c o n d i t io n s (6 3) d o n o t i m p o s e s i n g l e - v a l u e d r e s t r i c t i o n s o n t h e p o s i t io n o f t h e d i s c r e t e s p e c t r u m o f t h ec o r r e s p o n d i n g s p e c t r a l p r o b l e m a n d , f o r z ~ + ~ o t h e a s y m p t o t i c b e h a v i o r o f t h e k e r n e l K + m a y t u r n o u t t ob e e x p o n e n t i a l l y i n c r e a s i n g .

2 3 4


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T h i s d i f f i cu l t y d o e s no t a p p e a r f o r s c a l a r o p e r a t o r s a n d, a l s o , i f / ~ [ ~ 0 ] ~12 ~x ([1, ~2 ar e c on sta nts ) .O b v i o u sl y , t h e r e i s a l s o n o d i f fi c u l ty if th e c h a r a c t e r i s t i c v a l u e s i n t h e s p e c t r a l p r o b l e m f o r j = 2 a r ea b s e n t [ t h i s c a n h a p p e n i f t h e n o r m o f t h e i n i t i a l c o n d i t i o n s ~ i j ( x , y , 0) i s s u f f i c i e n t l y s m a l l ] . T h e m e t h o d sf o r o v e r c o m i n g t h e s e d i f f i c u l t i e s a s s o c i a t e d w i th t h e e x p o n e n t i a l g r o w t h o f t h e k e r n e l K + a n d c o r r e c tr e s u l t s r e l a t i v e t o th e C a u c h y p r o b l e m f o r c e r t a i n c l a s s e s o f i n t e g r a b l e s y s t e m s w i ll be th e s u b j e c t s o f t h es e c o n d p a r t o f o u r p a p e r .

L I T E R A T U R E C I T E D1 . C . S . G a r d n e r , J . M . G r e e n , M . K r u s k a l , a n d R . M . M i u r a , " M e t h o d f o r s o l v i ng t h e K o r t e w e g - d e V r i e sequ a t ion ," Phy s . R ev. Le t t . , 19._, 1095-1097 (1967) .2 . P . D . L a x , " I n t e g r a l s of n o n l i n e a r e q u a t i o n s o f e v o l u t io n a n d s o l i t a r y w a v e s , " C o m m . P u r e A p p l .

Ma th., 21._., 46 7- 49 0 (196 8).3 . V . E . Z a k h a r o v an d A . B . S h a b a t, " A r e f i n e d t h e o r y o f t w o - d i m e n s i o n a l s e l f - f o c u s s i n g a n d o n e - d i m e n -

s i o n a l s e l f - m o d u l a t i o n o f w a v e s i n n o n l i n e a r m e d i a , " Z h . l ~ k sp . T e o r . F i z . , 6 1 _, N o . 1, 1 1 8 -1 3 4 ( 1 9 7 1 ).4 . V . E . Z a k h a r o v a n d A . B . S h a b a t , " O n t h e i n t e r a c t i o n o f s o l i t o n s i n a s t a b l e m e d i u m , " Z h . ] ~ ks p . T e o r .

F iz ., 64.__, No. 5, 16 27 -16 39 (19 73) .5 . V . E . Z a k h a r o v , " T h e m e t h o d o f t he i n v e r s e s c a t t e r i n g p r o b l e m , " in : T h e o r y o f E l a s t i c M e d i a w i th

M i c r o s t r u c t u r e , e d i t e d b y I. A . K u k i n [ i n R u s s i a n ] , N a u k a , M o s c o w ( 1 97 4 ).6 . A . B . S h a b at , " C o n c e r n i n g t h e K o r t e w e g - d e V r i e s e q u a t i o n , " D ok l . A k a d. N au k S SS R, 2 1 1, No . 6 ,

1310-1313 (1973) .7 . V . E . Z a k h a r o v , " O n t h e p r o b l e m o f o n e - - d im e n s i o n al s t o c h a s t i z e d c i r c u i t s o f n o n l i n e a r o s c i l l a t o r s , "

Zh . t~ksp . T eo r. Fi z. , 65___, No. 1, 21 9-225 (1973).8 . B . B . K a d o m t s e v a n d V . I . P e t v i a s h v i l i , " O n t h e s t a b i li t y o f s o l i t a r y w a v e s in w e a k l y d i s p e r s i v e

me dia , " Dokl . Akad. Nauk SSSR, 192, No. 4 , 753-756 (1970).9 . M . W a d a t i a n d M . T o da , " E x a c t N -s o l it o n s o l u t io n o f t h e K o r t e w e g - d e V r i e s e q u a t i o n , " J . P h y s . S o c .

Ja pa n, 36.__, 140 3-14 12 (1972 ).1 0 . V . E . Z a k h a r o v an d S . V . M a n a k o v, " O n r e s o n a n c e i n t e r a c t i o n o f w a v e p a c k e t s i n n o n l i n e a r m e d i a , "

Z h ~ T F l>is. Re d. , 18___, No. 7, 41 3 (1973 ).1 1 . N . B l o e m b e r g e n , N o n l i n e a r O p t i c s, W . A . B e n j a m i n (1 9 65 ) .1 2 . V . E . Z a k h a r o v , L . A . T a k h t a d z h y a n , a n d L . D . F a d d e e v , " C o m p l e t e d e s c r i p t i o n o f t h e s o l u t io n s o f

t h e ' s i n - g o r d o n ' e q u a t i o n , " D o kl A k a d . N a uk S SS R (1 9 74 ) .1 3 . S . V . M a n a k o v , "O n a t h e o r y o f t h e t w o - d i m e n s i o n a l s t a t i o n a r y s e l f - fo c u s s i n g e l e c t r o m a g n e t i c w a v e s , "

Zh. ]~ksp . Teor . F iz . , 65_ 505-516 (1973) .1 4 . M . W a d a t i , " T h e m o d i f i e d K o r t e w e g - d e V r i e s e q u a t i o n , " J . P h y s . S o c . J a p a n , 34__, N o . 5 , 1 2 8 9 -1 2 9 1

(1973).

235

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