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Multi-soliton solutions of coupled nonlinear Schrödinger Equations

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1987 Chinese Phys. Lett. 4 185

(http://iopscience.iop.org/0256-307X/4/4/011)

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CHINESE PHYS. LETT. V o 1 . 4 , N o . 4 ( 1 9 8 7 )

CHINESE PHYSICS LETTERS S c i e n c e P r e s s , B e i j i n g

S p r i n g e r - V e r l a g

MULTI-SOLITON SOLUTIONS OF COUPLED N O N L I N E A R SCHRODINGER EQUATIONS

YAN Zhida ( I n s t i t u t e of A p p l i e d P h y s i c s & C o m p u t a t i o n a l M a t h e m a t i c s , B e i j i n g )

( R e c e i v e d 5 November 1 9 8 6 )

The 1+1 and l+l+l soliton solutions with different pola- rization for coupled nonlinear Schrodinger (NLS) Equations have been obtained by using Bilinear Method. behaviour and the polarization of these solutions are discus- sed.

The asymptotic

The exac t s o l u t i o n s o f t h e c o u p l e d or v e c t o r n o n l i n e a r S c h r o d i n g e r

e q u a t i o n have a t t r a c t e d much i n t e r e s t b e c a u s e of t h e i r a p p l i c a t i o n s

i p p r o b l e m s o f p l a s m a I n t h i s l e t t e r , w e show t h a t t h e c o u p l e d

or v e c t o r NLS e q u a t i o n c a n b e b i l i n e a r i z e d , a n d t h e m u l t i - s o l i t o n s o l u -

t i o n s of t h e s e e q u a t i o n s h a v e b e e n o b t a i n e d .

1. The Case of Two Coupled E q u a t i o n s : W e c o n s i d e r t h e s y s t e m of

e q u a t i o n s

w h i c h h a v e b e e n o b t a i n e d by many a u t h o r s i n v a r i o u s i n t e r e s t i n g p h y s i c s

problems!-4 F o r a s p e c i a l case A1=A = A , t h e s y s t e m (1.1) t u r n s t o t h e

v e c t o r NLS e q u a t i o n 2

5-7 which c a n a l s o b e o b t a i n e d f r o m Zakharov e q u a t i o n s . F o l l o w i n g H i r o t a ' s method,8 l e t + . = G . / F ( i = 1 , 2 ) , (1.1) c a n b e w r i t t e n

1 1 i n a b i l i n e a r form:

A A

(1.3) SG1*F=O , SG2*F=0 ,

D2F-F=-(A G G*+A G G*) , 1 X 2a 1 1 1 2 2 2

where F is r e a l , and t h e b i l i n e a r and S c h r o d i n g e r o p e r a t o r s are d e f i n e d A m n m n as S=iD + D 2 D D a - b = ( a - 3 ' 1 ( a t - a t I ) a ( x , t ) b ( x ' , t ' ) t x ' x t x x

i ( 1 . 4 a )

( 1 . 4 b )

The 1+1 s o l i t o n s o l u t i o n s : W e may w r i t e t h e e x p l i c i t forms of F a n d G

G l = e 5 ( 1 + C ( 5qr l*)eq+n*) , G 2 = e q ( 1 + C ( q < c * ) e 5 + ' * ) , F = l + e < + ' * C ( < 5 * ) + C ( ~ ~ * ) e n + ~ * + c ( c < * ~ q * ) e q+<+c. c .

w h e r e , q = P x - R t + n " , < = p x - w t + < " , a n d ' * I i m p l i e s a complex c o n j u g a t e . S u b s t i t u t i n g t h e e x p r e s s i o n s e q . ( 1 . 4 ) i n t o e q . ( 1 . 3 ) , it c a n b e shown

t h a t e q . ( 1 . 4 ) i s t h e s o l u t i o n of e q . ( 1 . 3 ) p r o v i d e d t h e f o l l o w i n g re la - t i o n s h o l d :

A A c ( < < * ) = 2 , / ( p + p * ) 2 , C ( n n * ) = ~ / ( P + P * ) 2 ,

186--- CHINESE PHYSICS LETTERS Vo1.4, No.4 c ( c c ) = ( p - p ) , c ( r , c * ) = l / ( p * + P ) , C ( i j k ) = C ( j k ) T C ( i n ) = C ( j k ) C ( i j ) C ( l k ) ,

. . . , C ( i j k ... m ) = C ( j k . . . m) fl C ( i n )

2 n ti

n fi id=ap2 ; l"T=ap2

where p , P , q o , a n d 5" are t h e a r b i t r a r y c o m p l e x c o n s t a n t s . H e n c e , t h e 1+1 s o l i t o n s o l u t i o n s w i t h f o u r i n d e p e n d e n t p a r a m e t e r s a r e

2. T h e Case of T h r e e C o u p l e d E q u a t i o n s : The r e s u l t s i n s e c t i o n 1 may

be g e n e r a l i z e d t o m u l t i - c o m p o n e n t S c h r o d i n g e r f i e l d s . For t h r e e - c o m p o - n e n t case, t h e s y s t e m of e q u a t i o n s is

v , v = l , 2 , 3 (2.1) 2 ( i a +aa 1 4 = - 4 2: A 1 4 ~ 1 2 , t x u v = l v

U s i n g t h e t r a n s f o r m a t i o n #J = G U / F , e q . ( 2 . 1 ) c a n be b i l i n e a r i z e d i n t o the f o l l o w i n g f o r m :

U

SG *F=O , u=1,2,3 I D ; F * F = z V = l ? A v G L ,G* \I - ( 2 . 2 ) 1-1 W e c a n y i e l d t h e s i m p l e s t n o n t r i v i a l s o l i t o n s o l u t i o n s f o r e q . ( 2 . 2 ) ,

which w i l l be cal led 1+1+1 s o l i t o n s o l u t i o n s :

vcp + ~ ( 1 1 * 2 2 * 3 3 * ) e x p ( ~ ~ ~ n ( ~ ) + C . c . )

where r- ( v ) = p ( V I x - R( v I t +p ( v ) a n d C ( v v * ) d e n o t e s C ( r , ( v ' q * ( ' ) ) , a n d so o n .

Q(') s a t i s f i e s t h e r e l a t i o n s i d v ) = a ( P ( u ) ) 2 ,

3 . A s y m p t o t i c Behaviour of t h e S o l u t i o n s : I n t h i s s e c t i o n , w e s h a l l p r o v e t h a t t h e s o l u t i o n s o b t a i n e d i n t h e p r e v i o u s s e c t i o n s s p l i t i n t o

t h e separa te s o l i t o n s i n t h e l i m i t of It1 + w , a n d a phase s h i f t of a s e p a r a t e s o l i t o n i n d u c e d b y c o l l i d i n g w i t h o t h e r s o l i t o n s i s o b t a i n e d .

F o r t h e s a k e of s i m p l i c i t y a n d c l e a r n e s s , w e s h a l l d i s c u s s o n l y t h e

1+1 s o l i t o n s o l u t i o n . A t f i r s t w e c o n s i d e r t h e case o f t + a . I n t h e l i m i t o f t + m , k e e p i n g 11 f i n i t e , f r o m e q . ( 1 . 5 ) w e f o u n d t h a t f o r R e (

cu-Rp/P)>O, R e ( p / P ) > O ,

lim$2=C-LI(nn*)e11m(q)sech(Re( n ) + 6 ; ) , 6 ;=1 /21nC( r n*) a n d k e -

4- y t e p i n g 5 f i n i t e , t h u s :

l i m 4 =c-' ( 5 <*)e i l m ( ' )+i61 sech(Re ( 5 ) +61) , 1 Q+

6;=1/2 I n ( c ( 5 s*n n* / C (q q* , A 1 = a r g ( p - P ) - a r g ( p + p * ) . The same p r o c e d u r e l eads t o t h e o t h e r a s y m p t o t i c f o r m f o r t - t - m . I n t h e l i m i t of t+-, k e e p i n g fi f i n i t e , t h e n

Vo1.4, No.4 Y A N Zhida 1 8 7

T h u s , w e have p r o v e d t h a t t h e s o l u t i o n s r e a l l y s p l i t i n t o s e p a r a t e

s o l i t o n s , a n d t h e c o l l i s i o n e f f e c t a p p e a r s o n l y i n t h e r e l a t i v e p h a s e

s h i f t s o f b o t h t h e e n v e l o p e a n d t h e p h a s e f a c t o r : F o r t h e e n v e l o p e ,

61=6+-6-=1/21n/C(Sn)121C(Sn*)j2, 6 2 = 6 z - 6 i = - 6 i ; and f o r t h e p h a s e f a c -

t o r , 6 =6 - 6 = 6 , 6 2 = % 2 - 6 2 = - % i . I t is c lear t h a t t h e t o t a l r e l a t i v e 1 d % + % - % + % + L

p h a s e s h i f t of t h e e n v e l o p e o r s o l i t o n s is z e r o ( 4 6 i = O ) , b u t t h e p h a s e

f a c t o r h a s n o n - z e r o t o t a l r e l a t i v e p h a s e s h i f t , Z d . = Z a r g ( P + p * ) .

4. The P o l a r i z a t i o n : The s o l u t i o n s of e q . ( l . l ) o r e q . ( 2 . 1 ) d e s c r i b e t h e i n t e r a c t i o n be tween S c h r o d i n g e r f i e l d s w i t h d i f f e r e n t p o l a r i z a t i o n s .

From t h e s o l u t i o n ( 1 . 5 ) , i t c a n b e c l e a r l y s e e n t h a t t h e i n t e r a c t i o n

be tween t h e s o l i t o n s w i t h d i f f e r e n t p o l a r i z a t i o n s i s s t r o n g , when t h e y

are close t o e a c h o t h e r . A s i m p l e s t s i t u a t i o n a r i ses , when P=p al lows a r e s o n a n t i n t e r a c t i o n be tween t h e t w o m o d u l a t e d wave t r a i n s . F o r t h i s

case, t h e two p o l a r i z e d s o l i t o n s h a v e t h e same v e l o c i t y a n d are bound

i n a compound s t a t e . P a r t i c u l a r l y , i f t h e r e l a t i o n c = j , + i $ is s a t i s f i e d

and b o t h s o l i t o n s are t r a n s v e r s e p o l a r i z a t i o n waves, a c i r c u l a r l y p o l a r i -

zed s o l i t o n w i l l be o b t a i n e d as f o l l o w s

f b 1

1 1

i I m ( n ) 8 a R e 2 ( p ) ) 4 , 6 = - l n A , $c=Asech(Re( T ) ) + 6 ) e ’ *=( A,+A, + L I

Where ++ a n d are l e f t - h a n d and r i g h t - h a n d c i r c u l a r l y - p o l a r i z e d s o l i - t o n s , which a re of o r t h o g o n a l s y s t e m . F o r a more g e n e r a l case of S = r i + @ , ( @ i s a n a r b i t r a r y complex c o n s t a n t e x c e p t 0 . i;, i r r , ) , t h e compound

s t a t e 4=+ e + Q 3 i s a n e l l i p t i c a l l y p o l a r i z e d s o l i t o n .

F i n a l l y , i t s h o u l d b e i n d i c a t e d t h a t t h e s o l i t o n s o l u t i o n s o b t a i n e d i n t h e p r e v i o u s s e c t i o n s c a n b e e x t e n d e d t o t h e N - s o l i t o n case f o r b o t h

t w o o r t h r e e and more components of S c h r o d i n g e r f i e l d . T h i s g e n e r a l i z a -

+ + l y 2 z

t i o n a n d i t s p r o o f w i l l b e discussed i n a n o t h e r p a p e r .

The a u t h o r w i s h e s t o t h a n k P r o f . GUO B o l i n g a n d P r o f . HE X i a n t u f o r many h e l p f u l a n d i n s t r u c t i v e d i s c u s s i o n s .

REFERENCES 1. 2 . 3 . 4 . 5 . 6 . 7 . 8 .

A.L.Berkhoer a n d V.E.Zakharov, JETP 3 1 ( 1 9 7 0 ) 4 8 6 . Y . I n o u e , 3 . P l a s m a P h y s . 1 6 ( 1 9 7 6 ) 4 3 9 . K.P.Das and S . S i h i , J . P l a s m a P h y s . 2 1 ( 1 9 7 9 ) 1 8 3 . M . R . Gupta e t a l . , J . P l a s m a P h y s . 2 5 ( 1 9 8 1 ) 4 9 9 ; 2 8 ( 1 9 8 2 ) 3 7 9 . V . E . Zakharov , JETP 3 5 ( 1 9 7 2 ) 9 0 8 . D . t e r Haar e t a l . , P h y s . R e p . 7 3 ( 1 9 8 1 ) 1 7 5 . HE X i a n t u , Acta P h y s . S i n . 3 2 ( 1 9 8 3 ) 3 2 5 ( i n C h i n e s e ) R . H i r o t a , J . M a t h . P h y s . 1 4 ( 1 9 7 3 ) 8 0 5 ; J . Phys.Soc.Jap.33(1972)14,56.