THE INSTABILITY OF ON-DIMENSIONAL LANGMUIR WAVE. SOLITONS AND COLLAPSE

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THE INSTABILITY OF ON-DIMENSIONAL LANGMUIR WAVE. SOLITONS AND COLLAPSE

N. Buchelnikova, E. Matochkin

To cite this version:

N. Buchelnikova, E. Matochkin. THE INSTABILITY OF ON-DIMENSIONAL LANGMUIR WAVE.

SOLITONS AND COLLAPSE. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-631-C7-632.

�10.1051/jphyscol:19797306�. �jpa-00219296�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6rnent au n07, Tome 40, JuiZZet 1979, page C7- 631

THE INSTABILITY OF M-DIMENSIONAL LANGMUIR WAVE. SOLITONS AND COLLAPSE

N.S. Buchelnikova and E.P. Matochkin.

I n s t i t u t e o f Nuclear Physics, Novosibirsk, U.S.S.R.

The numerical experiment was made t o i n v e s t i g a t e t h e i n s t a b i l i t y of onedimen

-

s i o n a l s t a n d i n g Langmuir waves w i t h amp-

E

2

l i t u d e s

w, = ^A

⁷⁷

(KO rd)

and phase 8 ~ n T

v e l o c i t i e s vp~/vT771 i n t h e c a s e s when plasma e l e c t r o n n o n l i n e a r i t y i s n e g l i g i b - l y small.

Case 1. W. = 4*10-*; v ~ h / ~ ~ = 1 6 ;

/

^{= 100. I n} t h i s c a s e t h e modulatio

-

rial i n s t a b i l i t y 1ead.s t o t h e f o r m a t i o n of two d e n s i t y c a v i t i e s and, t o t h e concen

-

t r a t i o n of t h e e l e c t r i c f i e l d . i n them.

A f t e r some time p r a c t i c a l l y a l l t h e f i e l d energy is c o n c e n t r a t e d J 1 t h e c a v i t i e s (Fig. 1-1, d o t t e d l i n e

-

t h e i n i t i a l d i s - t r i b u t i o n ) . T h i s s t r u c t u r e i s s t a b l e du- r i n g a l o n g time (Fig. 3,4). The change of e l e c t r i c f i e l d d i s t r i b u t i o n

t;)

du- r i n g plasma period Toe can be d e s c r i b e d by t h e formula f o r s t a n d i n g s o l i t o n s i m i - l a r t o t h a t f o r Langmuir s o l i t o n /I/:

-

= 0 , 5 Wh ;

bya=

^{27; = 0,27.}

So t h e modulational i n s t a b i l i t y i n t h i s c a s e l e a d s t o t h e f o r m a t i o n of q u a s i s t a b - l e s t a n d i n g s o l i t o n s .

Case 2. h, = 1 , 6 ; V ~ h / ~ - , = 1 6 ;

Xolrd

⁼ 100. I n t h i s c a s e w i t h h i g h e r i n i t i a l amplitude t h e modulational i n s t a - b i l i t y a l s o l e a d s t o e l e c t r i c f i e l d con- c e n t r a t i o n i n t h e c a v i t i e s (Fig. 2 ) and t o t h e f o r m a t i o n of s o l i t o n l i k e s t r u c t u r e (Fig. 1-2).

E L x , ~ )

d u r i n g plasma period changes l i k e t h a t of a s t a n d i n g s o l i t o n , but t h e parameters a r e nonequilibrium:

%

km i s h i g h e r and h/ho is l e s s t h a n t h o s e of a s o l i t o n . So t h e ponderomotive f o r c e must l e a d t o t h e f u r t h e r deepening of t h e c a v i t y . R e a l l y t h i s s t r u c t u r e i s u n s t a b l e and c o n t i n u e s t o c o n t r a c t up t o t h e beginning of t h e damping (Fig. 3). A t

f / ~ O e

-

1 2 t h e maximum energy d e n s i t y +

w m = 3,9; %Lo = 0 , 3 ; = 18;

L S X / A ~ = 0,18. The damping i s due t o t h e t r a p p i n g and a c c e l e r a t i o n of plasma e l e c - t r o n s by t h e s h o r t wavelength modes w i t h

- ^EL,

h /

- - ^{- =}

0 . 5 h 4 m low phase v e l o c i t i e s , It l e a d s t o t h e 8 7 h ^{7- J. t7c} f u l l a b s o r p t i o n of t h e f i e l d energy. Af- Both c u r v e s a r e shown on Fig. 2 ( d o t t e d

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

-

formula). The w i d t h AX a t t h e

c o n t i n u e t o i n c r e a s e due t o i o n i n e r t i a

,

t h e d e n s i t y p e r t u r b a t i o n

, l e v e l

.,.

^, (Fig. 4). A f t e r 1-2 i o n plasma p e r i o d s

i n t h e c a v i t y &/be a r e e q u a l t o t h o s e of

a t h e shock waves forme on t h e c a v i t y edgee

a s o l i t o n : W h c 6.10'~;

n/h.,

=

3*10'~

=

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797306

and t h e c a v i t y c o l l a p s e s ,

So i n t h i s c a s e we observe t h e pro- c e s s of c o l l a p s e /2,3/.

Case 3. Wo = I ,6; vph

/vT

= 48;

X o l r d = 300. I n t h i i c a s e

A,

and V P ~

a r e h i g h e r t h a n t h o s e i n t h e c a s e 2 and

0 we can e x p e c t t h a t t h e damping must begin iii

na

much l a t e r . R e a l l y t h e c o n t r a c t i o n of t h e s o l i t o n l i k e s t r u c t u r e i n t h i s c a s e c o n t i -

"'3Ha1B:H

^{0 50 I 100}

^o

^{50 2 ~ O O}

^o

^{ISO 3 300%}

nues f o r a l o n g e r time up t o

t/rOe.v

⁴¹

(Fig. 3 ) and l e a d s t o t h e h i g h e r energy

Fig. 1.

c o n c e n t r a t i o n and l o k a l i z a t i o n t h a n i n t h e

R,.--L-+

c a s e 2 (Pig. 1-3). I n t h i s c a s e t h e maxi-

*

mum energy d e n s i t y

w,

= 8,8; 0,3;

/

⁶ A x / A ~ = 0,055. The damping

SO SO 50 50 10Oy~

l i k e t h e c a s e 2 l e a d s t o t h e a b s o r p t i o n of

t h e f i e l d energy. Fig. 2,

So t h e m o d u l a t i o n a l i n s t a b i l i t y of

onedimensional Langmuir wave l e a d s t o t h e 5

-

^{W m G o} 3 f o r m a t i o n of s o l i t o n s i f wave amplitude

4 .

i s low enough WO ,( 3.10". I n o t h e r c a s e

c o l l a p s e p r o c e s s t a k e s p l a c e

-

t h e l o k a

-

^{3 .}

l i z a t i o n of e l e c t r i c f i e l d energy i n t h e

d e n s i t y c a v i t i e s , t h e c o n t r a c t i o n of t h e f c a v i t i e s and i n c r e a s e o f t h e energy den-

s i t y up t o t h e beginning o f t h e damping, 0 20

40 ^{6 0 8 0}

^to0

qGe

l e a d i n g t o t h e f u l l a b s o r p t i o n of e l e c t - r i c f i e l d energy by plasma e l e c t r o n s .

Fig. 3.

1.

L.

1,Rudakov Sov. Phys. Doklady

207,

821, 1972.

3

2. V.E.Zakharov Sov. Phys. JETP

62,

1745, 19720

3,

V.E.Zakharov, A.N.Rubenchik Sov, Phys.

JETP

65,

997. 1973.

f

0 20

40 60 80

f O O h o e Fig. 4.