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Submitted on 1 Jan 1979
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INTERACTION OF NONLINEAR ION-ACOUSTIC WAVES
C. Su
To cite this version:
C. Su. INTERACTION OF NONLINEAR ION-ACOUSTIC WAVES. Journal de Physique Colloques,
1979, 40 (C7), pp.C7-591-C7-592. �10.1051/jphyscol:19797286�. �jpa-00219274�
JOURNAL DE PHYSIQUE CoZloque
C7,suppZ6ment au n07, Tome
40,JuiZZet
1979,page C7- 591
INTERACTION OF NONLINEAR ION-ACOUSTIC WAVES
C.H.
Su.
Division o f Applied Mathematics, Brown University, Providence,
R . I . 02912, U.S.A.We s t u d y t h e head-on c o l l i s i o n s be- ( 4 ) , ( 5 ) makes t h e n o n - n e u t r a l i t y c o r r e c - tween two s o l i t a r y waves i n a c o l d plasma t i o n coming- i n t o p r e s e n c e same a s t h a t o f d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n s : n o n l i n e a r i t y .
Cand
CLa r e wave s p e e d s ,
R
nt +
=0, (1)
8and I$ phase f u n c t i o n s f o r t h e r i g h t - and u
2ut + ( 2 - + q J ) x
=0 , ( 2 ) l e f t - g o i n g waves r e s p e c t i v e l y - I n t h e
$,, = n - e t ( 3 second o r d e r a p p r o x i m a t i o n i n
E,we o b t a i n where n , u d e n o t e d e n s i t y and v e l o c i k y o f
p o s i t i v e i o n s ,
)Ia dimens,ionless p o t e n t i a l . . E q u a t i o n ( 3 ) , t h e P o i s s o n equation,, , k d i r c a t e s t h a t t h e e l e c t r o n s a r e - t a k e n t o b a i n t h e r m a l equiLihrium.. .The' e x i s t e n c e o f s o l i t a r y waves 2n t h i s system h a s been d e m o n s t r a t e d by Sagdeev (1966), Su and Gardner ( 1 9 6 9 ) - The catch-up c a l l i s i o n of a t a l l e r wave t o a s m a l l e r . w a v e c a n be i n - f e r r e d from t h e e l e g a n t . r e s u l t f o r -the Korteweg-devries e q u a t i o n -done.,by
G a x d n e r ,Green, K r u s k a l and. Miura ( 1 9 6 7 ) . Here we s t u d y head-on c o l l i s i o n s by a p e r t u r b a t i ~ n method making assump.tien of long wave ( a l - most q u a s a i - n e u t r a l i t y ) and s m a l l . a m p l i - t u d e s of t h e c o l l i d i n g wLves. I n t h e f i r s t o r d e r a p p r o x i m a t i o n we . o b t a i n t h a k t h e c o l l i d i n g waves have t h e i r . w a v e f i e u s s a t i s f y f n g t h e Korteweg-de'ilrkes eq.uation, i . e . a s ( < ) and b S ( g ) where S ( 5 )
=s e c h g,
a , b a r e c o n s t a n t s . We h a v e - u s e d t h e wave-
(1) Q u a d r a t i c t e r m s i n S a r e added t o t h e l i n e a r . one.
( 2 ) L i n e a r i o n - a c c o u s t i c wave-speeds a r e m o d i f i e d by t e r m s which depend on wave a m p l i t u d e s .
( 3 ) The l o w e s t o r d e r p h a s e . f u n c t i o n s
e(O) (17) f o r I $ ( O ) ( 5 ) 1 a r e a f u n c t i o n o f n ( o r 5 ) a l o n e . These f u n c t i o n s change by a c o n s t a n t v a l u e a s t h e i r argument v a r i e s o v e r a n i n f i n i t y . A f t e r t h e waves s u f f e r a c o l l i s i o n ,
t h e y have a c o n s t a n t amount o f p h a s e s h i f t . The s i g n s of t h e s e p h a s e s h i f t s make. t h e a r r i v a l of t h e waves l a t e r t h a n t h e y s h o u l d .
The wave f i e l d s o f e a c h wave a f t e r . t h e i r c o l l i s i o n i s i d e n t i c a l t o t h a t b e f o r e ,
.-' .
i . e . t h e waves p r e s e r v e t h e i r i a e n t i t i e s . I n t h e t h i r d o r d e r o f appr.oximation, t h e same t h r e e c o r r e c t i o n s ' above show up(
w i t h . t h e quadratic. tern i n . (1). r e p l a c e d framed c o o r d i n a k a s y s t e m by a c u b i c polynomials- i n S. However, a
5 =&k(x-c,t) + ~ k e ( S , r l ) ( 4 ) s a l i e n t p r o p e r t y o f t h e p h a s e f u n c t i o n s
n = & R (x+CLt) + E R G ( E r n ) ( 5 ) comes i n t o p l a c e , i n t h a t e(') and @ ( l ) where
Ei s a d f m e n s i o n l e s s parameter f o r a r e f u n c t i o n s o f b o t h 5 a n d . Emerging
s m a l l a m p l i t u d e , t h e u s e of.
E'''i n Eqs. from t h e i r c o l l i s i o n , b o t h waves-now w i l l 39
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797286
have a p h a s e - s h i f
twhich i s d i f f e r e n t a t d i f f e r e n t p o i n t s of t h e wave. The wave p r o f i l e s a r e t h u s d i s t o r t e d from what were b e f o r e t h e c o l l i s o n . Since t h i s d i s - t o r t e d wave f i e l d does n o t s a t i s f y t h e e q u a t i o n f o r waves which a r e propagated.
w i t h o u t change i n s h a p e and speed. We s t u d y t h e slow-time e v o l u t i a n . of-.
. t h o s ewave f i e l d s immediately a f t e r t h e . a o l l i - s i o n . I t i s f.ound t h a t e a c h of t h e s e waves s p l i t s i n t o two p a r t s : T h e m a i n p a r t h a s e x a c t l y the. same . p r o f i l e and wave v e l o c i t y as t h e one b e f o r e c o l l i s i o n
( a p a r t .with a c0nstant.phas.e .shift).
.The.o t h e r p a r t h a s a shape of L e t t e r N h i g h l y s t r e t c h e d o u t i n horizaata-l d i x e n k b n . These secondary. .waveLets .propagate w i t 2 1 diminishing amplitude, i n t h e o p p o s i t e d i r e c t i o n t o t h e propagation of t h e main wave.
Our main c o n c l u s i o n a b o u t t h e e f f e c t of a head-on c o t l i s i o n between two s a l i - t a r y waves a r e
1) Constant phase s h i f t s , . The wav.es g e t r e t a r d e d d u r i n g t h d r c o l E i s i o n . How- e v e r , each wave recoxer-s i t s speed and shape a f t e r c o l l i s i o n .
2 )
