STABILITY OF THE NONLINEAR PERIODIC WAVES IN PLASMA

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STABILITY OF THE NONLINEAR PERIODIC WAVES IN PLASMA

V. Pavlenko, V. Petviashvili

To cite this version:

V. Pavlenko, V. Petviashvili. STABILITY OF THE NONLINEAR PERIODIC WAVES IN PLASMA.

Journal de Physique Colloques, 1979, 40 (C7), pp.C7-621-C7-622. �10.1051/jphyscol:19797301�. �jpa- 00219291�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, page C7- 6 2 1

STABILITY OF THE NONLINEAR PERIODIC WAVES IN PLASMA

V.P. P a v l e n k o a n d V.I. Petviashvili.

~izs t i t u t e f o r ~ u c Zear Research, Kiev, U. S. S. R.

It was announced i n paper /1/ about l e a d s t o t h e following d i m e n s i o n l e s ~ eq observing of 'chains of amplitude modula-

t e d nonlinear Langmuir waves with quasi periodic space s t r u c t u r e accompanied by corresponding d e n s i t y p i t s . From t h i s we conclude t h a t t h e amplitude modulation weakenes considerably i n s t a b i l i t y of mo- nochromatic Ungmuir wave. Waves l i k e the observed ones i n /1/ can e x i s t a l s o i n p l a s m containing devices with condition

and i n awroral region of t h e mag- nitosphere where they a r e responsible f o r t h e anomalous r e s i s t i v i t y .

Usually t h e problem of s t a b i l i t y of periodic wave r e l a t i v e t o i n f i n i t l y smal.1 perturbations i s reduced t o f i n d i n g of

( 7 )

2

?

X i s dimensionless coordinate,

n

21

^is

wanted increment, &,)a t h e periodic so- l u t i o n of t h e ISE t h a t i s expressed thro- ugh e l l e p t i c a l Jacoby functions Ca o r dn I n dimensionalless case

(

depends only on X and '2

,

^{where 2} is t h e modulus of e l l e p t i c a l functions. We expand

4

^over

~ K x

loch func tons

\P,

where

9,

(x) =

e _uM

_fx),

KO is t h e main wave number of periodic- 2

a 1 wave ^f0

.

A f t e r t h i s t h e problem i s reduced t o f i n d i n g of eigenvalues of t h e i n f i n i t e matrix

eigenvaluee of l i n e a r d i f f e r e n t i a l eq.

with p e r i o d i c a l c o e f f i c i e n t s . I n preceed-

$

^{Ma. =}

[I+ d(n+~)fl'd!,~ +3I, -

⁽²⁾

0

-

i n g s t u d i e s these perturbations were ex-

- {4

⁺ '.K

b

(&+N)z+ ( m + ~ ) < ] j

1,

pended i n unadequate system of basic fun-

c t i o n s t h a t sometiaes l e d t o error. I n ~ K / K o

prersent paper we expand t h e perturbation

i n i n t e g r a l over quasiimpulses on Bloch I f 2 - i s not too c l o s e t o 1-the f u n c t i - f u n c t i o n s t h e only complete basic eystem ons

I

^and

1,

decrease with s u f f i c i e n t

1

f o r eq. with periodic dependence of pa- r a t e with growth of

I

m - n l . Therefore i t rameters on coordinat s. i s p o s s i b l e t o r e s t r i c t ourselves with

Let us consider t h e , n o n l i n e a r consideration of f i n i t e matrix 25*25. We Shredinger eq.(l?~E). The s t a b i l i t y pro- considered t h i s matrix i n t h e region blem of periodic s o l u t i o n s of t h i s eq. O.$

ae

4 4 0.999. The a n a l i s i s of computed

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797301

data reveals the existence the instability region. The dependence of maximal incre- ment on 2 is given in following table

The more is 2 the deeper is amplitude modulation. The according variation of

energy density is small. We see that the growth of amplitude modulation causes de- creasing of instability increment to the value which makes possible the existence of wave chains observed in /I/. Note that the known result on instability of perio- dic Langmuir wave corresponds to the case

z

^{so /2/.}

The similar investigation of stability of periodic solutions of KdV eq. shows that solutions are always stable. 'Pbe re- sult is in accordance with the nonlinear consideration made in /3/ with help of the method of the reversed problem of scattering.

The KdV eq. generalized for two dim- ensional case has the form /4/

The one dimensional solutions of this eq.

2 2

are expressed through Cn or dn

.

^{Let us}

consider the stability of these solutions relative to two dimensional perturbati- ons of the form

'$1

⁺ g t

4

^{(%.I)= 'P(x) e}

Diagonalisation of this matrix shows that all solutions are stable in case of negative dispersion (the lower sign in rhs of eq ( 3 ) ) . If the dispersion is posi- kive all one dimensional solutions are unstable. The dependence of maximal inc- rement on 2 is given in following table

Note that when K 2 = O or for big K the

2

the solutions are stable.

The considered examples illustrate the proposed method for solving of stabi- lity problem of periodic solutions of other nonlinear eqe. including the solu- tions consisting of solitone.

It is interesting that this method gives simple way of getting zonal pictu- re of eigenfrequences of medium in pre- sence of the periodic wave.

Ref erenc ea

1. Antipov S.V.,Keslin M.V., Sneehkin E.

Trubnikov A.S. Zh. exsp. teor. fiz.

14,

965, 1978.

2. Zakharov V.E. Zh. exsp. teor.fiz.62, 1745, 1972.

3. Kusnetsov E.A.,Mi.khailov A.V. Zh.

exsp. teor. fis.

3,

1717,197.4.

4. Kadomtsev B.B. Petviaehvili V.I.

D0kl.M SSSR mpanding (P

($1

in the integral over

Bloch functions after some transformati- ons we get the matrix