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Multidimensional and dissipative solitons
Physic~ 3D (1981) 1 & 2,329-$34 © North-HoUandPublishing Company
M U L T I D I M E N S I O N A L AND D I S S I P A T I V E Vladimir I.V.
SOLITONS
I. P e t v i a s h v i l i
K u r c h a t o v I n s t i t u t e of A t o m i c Moscow, 123182, U.S.S.R.
Energy
S o l u t i o n s of h i d r o d i n a m i c and p l a s m a e q u a t i o n s in u n s t a b l e cases are found in form of 2 or 3 d i m e n s i o n a l s o l i t a r y w a v e s . A m p l i t u d e s and d i m e n s i o n s of them are d e t e r m i n e d by the b a l a n c e in w a v e v e k t o r space b e t w e e n the e n e r g y input in i n s t a b i l i t y r e g i o n and nonl i n e a r t r a n s p o r t w i t h c o n s e q u e n t diss i p a t i o n in range to big w a v e n u m b e r s . The s o l u t i o n s are found with help of simple n u m e r i c a l m e t h o d of s t a b i l i z i n g factor
When s t u d i n g the o s c i l l a t i o n s d e v e l o p i n g in n o n l i n e a r d i s p e r s i v e m e d i a , w e u s u a l l y use s i m p l i f i e d e q u a t i o n s d e r i v e d from general relations d e s c r i b i n g the medium. But even after m a x i m a l s i m p l i f i c a t i o n we often obtain e q u a t i o n s w h i c h are d i f f i c u l t for s o l u t i o n or i n v e s t i gation. In many cases this d i f f i c u l t y may be o v e r c o m e by using the m e t h o d of s t a b i l i s i n g factor If]which allows to o b t a i n n u m e r i c a l l y s t a t i o n a r y s o l u t i o n s of n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s with high o r d e r of d e r i v a t i v e s in form of m u l t i d i m e n s i o n a l p e r i o d i c a l or s o l i t a r y w a v e s . B y this m e t h o d the s o l i t o n s o l u t i o n s of t w o d i m e n s i o nal KdV e q . , d e r i v e d i n [ 2 ] , a n d of L a n g m u i r wave in weak m a g n e t i c field e q u a t i o n were obtained. R e c e n t l y t w o d i m e n s i o n a l KdV solitons were o b t a i n e d a n a l i t i c a l l y [ ~ I These are e x a m p l e s of " e l a s t i c " s o l i t o n s in w h i c h the s i n c h r o n i s m for all w a v e v e k t o r modes c o n s t i t u t i n g s o l i t o n is c o n t a i n e d by the elastic n o n l i n e a r f o r c e s , w h i c h o v e r c o m e d e s i n c h r o n i s a t i o n t r e n d c a u s e d by the d i s p e r s i o n . In u n s t a b l e m e d i u m s o l i t o n s are g r o w i n g untill e n e r g y input in i n s t a b i l i t y region in wave v e c t o r space is b a l a n c e d by the n o n l i n e a r t r a n s p o r t and c o n s e q u e n t d i s s i p a t i o n in region of large w a v e vectors, then s o l i t o n s b e c o m e s t a t i o n a r y ( d i s s i p a t i v e ) . D i s s i p a t i v e s o l i t o n s are similar. M e t h o d of s t a b . f a c t o r allows to i n v e s t i g a t e d i s s i p a t i v e s o l i t o n s too and check the p r i n c i p a l d e p e n d e n c e of p a r a m e t e r s of t h e m on i n s t a b i l i t y i n c r e a m e n t . C o n s i d e r the solitons on the viscous l i q u i d film f l o w i n g down on the wall [5] (Fig.l) the i n s t a t i l i t y of such flow was found i n ~ ] . The s i m p l i f i e d d i m e n s i o n l e s s e q u a t i o n of such flow is
+
+
+ 329
]
=o
(1)
330
Figure
V.I. Petviashvili / Multidimensional and dissipative solitons
l : Solitons (upper).
on the v i s c o u s l i q u i d f i l m f l o w i n g M a t h e m a t i c a l m o d e l (below).
down
the w a l l
I/.L Petviashvili /Multidimensional and dissipative solitons
331
Fourth d i s s i p a t i v e term is due to surface t e n s i o n , s e c o n d n o n l i n e a r term describes the t r a n s p o r t of energy from small wave numbers to large ones from u n s t a b i l i t y to dissipation. On Fig.1 s o l u t i o n of this e q u a t i o n ~ = ~ - ¢ ~ j Y J i s shown. The s o l u t i o n e~ists only w h e n soliton velocity is c = O . 6 2 1 1 , a n d then soliton a m p l i t u d e i s A ~ a ~ 0.726. Even in one d i m e n s i o n a l case when order of d e r i v a t i v e s is more than 2 , s o l i t o n s o l u t i o n s are easier to obtain by the s t a b . f a c t o r method, For example take one d i m e n s i o n a l d e s c r i b e d by the eq.:
a, 3 +
soliton
solution
of
(1),which
-cL=o,
is
(2) C which
This p r o b l e m has three e i g e n v a l u e s . A m o n g them is found by O.Yu. T s v e l o d u b is equal to 1.2161.
as was
We illustrate the m e t h o d c o n s i d e r i n g the d i s s i p a t i v e s p h e r i c a l l y symm e t r i c L a n g m u i r soliton. In many e x p e r i m e n t s on i n t e r a c t i o n of monochomatic e.m.wave with plasma it was observed that in plasma the cavitations of d e n s i t y (cavitons) appear with c h a r a c t e r i s t i c size of order of few Debye lengths. The lifetime of cavitons was much larger than c h a r a c t e r i s t i c time of collapse. The spectral analysis of e . m . w a v e s scattered in these e x p e r i m e n t s show that very n a r r o w e n e r g e t i c spectral lines were located at f r e q u e n c i e s near 2~ and 3/2~ where is f r e q u e n c y of incident wave. These facts can be e x p l a i n e d by the creation of p a r a m e t r i c a l l y fed d i s s i p a t i v e solitons which o r i g i n a t e at places where feeding is e f f e c t i v e n a m e l y where local plasma frequency is equal to ~ or ~/2. The i n t e r a c t i o n of incident wave with these cavitons g e n e r a t e s scattered r a d i a t i o n of waves with combination f r e q u e n c i e s 2~, 3/2~. The linear part of s i m p l i f i e d e q u a t i o n of L a n g m u i r waves c o r r e s p o n d s to linear d i s p e r s i o n equation:
oO=Wp (
~
~
K~÷g
-ZA
r i s i n c r i m e n t , /~ is L a n d a u damping. If we take the n o n l i n e a r term as in Z a k h a r o v
K)
(3)
eq. 7 then we obtain:
R z-~a
8E +.&RE_2R~E+IEI~E 2 + z r E - 6ze
i a7
R , 03
time
t is in units 6 0 ; ¢, ~ is in Debye length u n i t s , E ,p is p l a s m a p r e s s u r e . L a n d a u damping we take is i n c r e m e n t due to p a r a m e t r i c feeding,
(~)
in units in model
form
In contrast w i t h s t a t i o n a r y t u r b u l e n c e we have only one f r e q u e n c y d i s s i p a t i v e s o l i t o n , a n d this helps with l o c a l i s a t i o n in space,In s t a t i o n a r y case we have: a (~)
6.g...~
, Ig=I 1%< 1,
n
in
<< ¢
f here is complex funct£on,/~, is real. Complete f r e q u e n c y in the caviton is~)p(~-~C4).Equation (4) is v a l i d w h e n i n e q u a l i t i s in (5) are satisfied. S u b s t i t u t i n g (5) in (4) we obtain o r d i n a r y differential eq.with ~ o u n d a r y c o n d i t i o n s f ( 0 ) ~ f(oo)= O. _O. is e i g e n v a l u e d e p e n d i n g on r .This e q u a t i o n with help of l i n e a r Green's function takes the form of n o n l i n e a r integral equation.
V.I. Petviashvili /Multidimensional and dissipative solitons
332
o
(6)
~']~'z
-,
-,
G~ = ~ + } K % 2 K C
,/a
ZF~ +
~SZk'°
(8)
- is B e s s e ! f u n c t i o n , O~- is F u r i e r - B e s s e l s p e c t r u m of l i n e a r oper a t o r in ( h ) . E q . ( 6 ) has s o l i t o n s o l u t i o n as far as i n t e g r a l (7) is regular. I t e r a t i o n m e t h o d a p p l a i e d to (6) gives d i v e r g i n g s e q u e n c e of aprox i m a t i o n s . T o m a k e the s e q u e n c e c o n v e r g i n g we i n t r o d u c e the s t a b . f a c tor s: ~ ~
o
When
f is s o l u t i o n
@
of
(6) we
have:
s = l. So we
instead
of
(6)
solve
R i g h t h a n d side of (i0) - has d e g r e e of n o n l i n e a r i t i e q u a l to O. W h e n we solve (i0) by i t e r a t i o n s d i v e r g i n g e n c e d i s a p e a r s but the se~ q u e n c e is not c o n v e r g i n g u n t i l we find c o r r e c t e i g e n v a l u e ~ l .When we s e l e c t p r o p e r e i ~ e u v a l u e the s e r i e s b e c o m e q u i c k l y c o n v e r g i n g , and at the same time s tends to i. In f o l l o w i n g t a b l e the e i g e n v a l u e s . ~ , are given efficients in s p e c t r u m of i n c r e m e n t /~ in cases 2 .lO -4
1.5.1o -3
corresponding whenC-a(#l~Km
to co2K#I#~l
0.oo3
o,o0~
0,Ol
o,o16
0,23
~(i~ 1 . 4 . 1 0 -3 7.10 -3
0,01
0,014
0,028
O,Oh
O,O56
~3
0,02
0,06
0,08
0,I
0,16
0,2
/~-
0,027
O,O57
0,08
0 I
0,14
0,17
~
~,o,..-
~
js9
Q ego Figure
~ol
0 f
I
,
gd
2.
D e p e n d e n c e of e l e c t r i c f i e l d E i n ~ - - ~ u n i t s s o l i t o n centre in D e b y e radius u n i t s . Figure
0,2
3.
Deoendence of maximal dencity of energy ing distance from s o l i t o n c e n t r e -~ _ and on r e l a t m v e f r e q u e n c y of solmton.~..
on
!
distance
from
= , E ~! of corresponds o l i t o n e n e r g y $ -- ~?TSIE%Itdz
V.[. Petviashvili / Multidimensional and dissipative solitons
333
O t h e r r e s u l t s are g i v e n on f i g u r e s 2 , a n d 3 . L i k e s o l i t o n s on v i s c o u s l i ~ u l d film c a v i t o n s are r a n d o m l i d i s t r i b u t e d in u n s t a b l e p l a s m a . L i f e t i m e of c a v i t o n s is d e t e r m i n e d by c o l l i s i o n s w i t h n e i g h b o u r e s . C a v i t o n t u r b u l e n c e is c h a r a c t e r i s e d by d i s c r e t e s p e c t r a l lines of f r e q u e n c l e s , w h i l e t u r b u l e n t or c o l l a p s i n g L a n g m u i r o s c i l l a t i o n s p o s ses w i d e f r e q u e n c y s p e c t r u m . In the s t e l l a r a t o r s in r e g i m e s w h e n ~ p < ~ ¢ a n d w i t h l o n g i t u d i n a l e l e c t ! ric f i e l d less than D r e i s e r s p o w e r f u l l l a n g m u i r o s c i l l a t i o n s l o c a l i s e d on s m a l l r a d i u s and on p o l o i d a l a n g l e w e r e o b s e r v e d . The l o c a l i s a t i o n r a n g e c o i n c i d e d w i t h r u n a w a y e l e c t r o n s w h i c h i n d i c a t e s that l a s t ones w e r e f e e d i n g L a n g m u i r w a v e s o l i t o n s a p p a r e n t l y by the anomalous Dopier effect . The s p e c t r u m of e x c i t e d w a v e s was very n a r r o w . The h i g h d e g r e e of m o n o c h r o m a t i c i t y and l o c a l i s a t i o n of this w a v e s can not be e x p l a i n e d by the w e a k t u r b u l e n c e t h e o r y . We s h o w that in this case the e x i s t e n c e of L a n g m u i r w a v e s o l i t o n p e r i o dical a l o n g m a g n e t i c f i e l d and l o c a l i s e d a c r o s s the one is p o s s i b l e . The d e n s i t y w e l l can not a r i s e in the p r e s e n c e of r u n a w a y e l e c t r o n s ( t a i l e on the e l e c t r o n d i s t r i b u t i o n f u n c t i o n ) b e c a u s e the w a v e s p r o p a g a t i n g r e v e r s e l y to the r u n a w a y e l e c t r o n s are d u m p e d by the n o r m a l D o p p l e r e f f e c t . S o o n l y n o n l i n e a r i t i e s of w a v e s p r o p a g a t i o n g in tale d i r e c t i o n s t a y acting. T h e n e q . o f s o l i t o n in w h i c h d e p e n d e n c e on coo r d i n a t e alon 9 m a g n e t i c f i e l d z and t i m e t o c c u r o n l y in c o m b i n a t i o n ~-~p U ~ ~ t a k e s the form: ~ 2e~
Here ~ i s
electric
potential,angular
brackets
!fl~|~< 4 , a f t e r e x p a n d i n g (ii) in p o w e r s o l u t i o n in the form of p l a n e s o l i t o n :
denote
series
of~
averaging we
by Z.
easily
find
~ h e n r u n a w a y e l e c t r o n s p u m p the s o l i t o n its a m p l i t u d e A g r o w s and a d i t i o n a l h a r m o n i c s a r i s e t i l l the h a r m o n i c w i t h w a v e l e n g t h of order of D e b y e l e n g t h g r o w s s u f f i c i e n t l y large. Th~ soliton becomes dissipative with amplitude~%and eq.(I) is to be s o l v e d by the m e t h o d of stab factor. The a x i a l l y s i m m e t r i c s o l u t i on can be o b t a i n e d . D r i f t w a v e s also f o r m a x i a l l y s y m m e t r i c s o l i t o n s but they are i n c l i n e d by small a n g l e to d i r e c t i o n of m a g n e t i c f i e l d and p r o p a g a t e w i t h v e l o c i t y m o r e than e l e c t r i c d r i f t one. The m o s t i m p o r t a n t n o n l i n e a r i t y for d r i f t w a v e s s o l u t i o n is c a u s e d by e l e c t r o n t e m p e r a t u r e g r a d i e n t . I o n i c or e l e c t r o n i c c i c l o t r o n w a v e p a c k e t s g e n e r a t e d i a m a g n e t i c c u r r e n t due to hf p r e s s u r e g r a d i e n t across the m a g n e t i c f i e l d , w h i c h c r e a t e s m a g n e t i c w e l l in sites of w a v e p a c k e t s [ 9 1 The f r e q u e n c y of c i c l o t r o n w a v e s is as s e n s i t i v e to m a g n e t i c f i e l d v a r i a t i o n s as L a n g m u e r w a v e f r e q u e n c y to d e n c i t y v a r i a t i o n s . S o like L a n g m u i e r w a v e s c i c l o t r o n w a v e s e l f l o c a l i s a t i o n is p o s s i b l e . F o u n d the r e l a t i o n b e t w e e n w a v e e n e r g y d e n c i t y and m a g n e t i n g w e l l a m p l i t u d e it is e a s y to c a l c u l a t e p a r a m e t e r s of c i c l o t r o n s o l i t o n s [~.
REFERENCES i. P e t v i a s h v i l i V . l . , F i s i k a P l a s m i , 2 ( 1 9 7 6 ) 469 2. K a d o m t s e v B . B . , P e t v i a s h v i l i V . I . D o k l a d i Akad. N a u k 753 3. P e t v i a s h v i l i V.I. F i s i k a P l a s m i , l (1975) 28
SSSR,192
(1970)
334
V.L Petviashvili / Multidimensional and dissipative solitons
h. Manakov S.V.,Zakharov V.E.,Bordag L.A.,Its A.R.,Matveev V.B., Phisics Lett.,63A (1977) 205 5. Petvlashvili V.I.,Tsvelodub O.Yu.,Dokladi Aead. Nauk SSSR,238 (1978) 1321 6. Benjamin T.B.,Fluld. Mech.,2 (1957) 55h 7. Zakharov V.E.,JETP,62 (1972) 17h5 8. Petvlshvili V.I., Fisika Plasmi,3 (1977) 270 9. Nekrasov A.K.,Petviashvi!i V.I.,JETP,77 (1979) 605.
M U L T I D I M E N S I O N A L AND D I S S I P A T I V E Vladimir I.V.
SOLITONS
I. P e t v i a s h v i l i
K u r c h a t o v I n s t i t u t e of A t o m i c Moscow, 123182, U.S.S.R.
Energy
S o l u t i o n s of h i d r o d i n a m i c and p l a s m a e q u a t i o n s in u n s t a b l e cases are found in form of 2 or 3 d i m e n s i o n a l s o l i t a r y w a v e s . A m p l i t u d e s and d i m e n s i o n s of them are d e t e r m i n e d by the b a l a n c e in w a v e v e k t o r space b e t w e e n the e n e r g y input in i n s t a b i l i t y r e g i o n and nonl i n e a r t r a n s p o r t w i t h c o n s e q u e n t diss i p a t i o n in range to big w a v e n u m b e r s . The s o l u t i o n s are found with help of simple n u m e r i c a l m e t h o d of s t a b i l i z i n g factor
When s t u d i n g the o s c i l l a t i o n s d e v e l o p i n g in n o n l i n e a r d i s p e r s i v e m e d i a , w e u s u a l l y use s i m p l i f i e d e q u a t i o n s d e r i v e d from general relations d e s c r i b i n g the medium. But even after m a x i m a l s i m p l i f i c a t i o n we often obtain e q u a t i o n s w h i c h are d i f f i c u l t for s o l u t i o n or i n v e s t i gation. In many cases this d i f f i c u l t y may be o v e r c o m e by using the m e t h o d of s t a b i l i s i n g factor If]which allows to o b t a i n n u m e r i c a l l y s t a t i o n a r y s o l u t i o n s of n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s with high o r d e r of d e r i v a t i v e s in form of m u l t i d i m e n s i o n a l p e r i o d i c a l or s o l i t a r y w a v e s . B y this m e t h o d the s o l i t o n s o l u t i o n s of t w o d i m e n s i o nal KdV e q . , d e r i v e d i n [ 2 ] , a n d of L a n g m u i r wave in weak m a g n e t i c field e q u a t i o n were obtained. R e c e n t l y t w o d i m e n s i o n a l KdV solitons were o b t a i n e d a n a l i t i c a l l y [ ~ I These are e x a m p l e s of " e l a s t i c " s o l i t o n s in w h i c h the s i n c h r o n i s m for all w a v e v e k t o r modes c o n s t i t u t i n g s o l i t o n is c o n t a i n e d by the elastic n o n l i n e a r f o r c e s , w h i c h o v e r c o m e d e s i n c h r o n i s a t i o n t r e n d c a u s e d by the d i s p e r s i o n . In u n s t a b l e m e d i u m s o l i t o n s are g r o w i n g untill e n e r g y input in i n s t a b i l i t y region in wave v e c t o r space is b a l a n c e d by the n o n l i n e a r t r a n s p o r t and c o n s e q u e n t d i s s i p a t i o n in region of large w a v e vectors, then s o l i t o n s b e c o m e s t a t i o n a r y ( d i s s i p a t i v e ) . D i s s i p a t i v e s o l i t o n s are similar. M e t h o d of s t a b . f a c t o r allows to i n v e s t i g a t e d i s s i p a t i v e s o l i t o n s too and check the p r i n c i p a l d e p e n d e n c e of p a r a m e t e r s of t h e m on i n s t a b i l i t y i n c r e a m e n t . C o n s i d e r the solitons on the viscous l i q u i d film f l o w i n g down on the wall [5] (Fig.l) the i n s t a t i l i t y of such flow was found i n ~ ] . The s i m p l i f i e d d i m e n s i o n l e s s e q u a t i o n of such flow is
+
+
+ 329
]
=o
(1)
330
Figure
V.I. Petviashvili / Multidimensional and dissipative solitons
l : Solitons (upper).
on the v i s c o u s l i q u i d f i l m f l o w i n g M a t h e m a t i c a l m o d e l (below).
down
the w a l l
I/.L Petviashvili /Multidimensional and dissipative solitons
331
Fourth d i s s i p a t i v e term is due to surface t e n s i o n , s e c o n d n o n l i n e a r term describes the t r a n s p o r t of energy from small wave numbers to large ones from u n s t a b i l i t y to dissipation. On Fig.1 s o l u t i o n of this e q u a t i o n ~ = ~ - ¢ ~ j Y J i s shown. The s o l u t i o n e~ists only w h e n soliton velocity is c = O . 6 2 1 1 , a n d then soliton a m p l i t u d e i s A ~ a ~ 0.726. Even in one d i m e n s i o n a l case when order of d e r i v a t i v e s is more than 2 , s o l i t o n s o l u t i o n s are easier to obtain by the s t a b . f a c t o r method, For example take one d i m e n s i o n a l d e s c r i b e d by the eq.:
a, 3 +
soliton
solution
of
(1),which
-cL=o,
is
(2) C which
This p r o b l e m has three e i g e n v a l u e s . A m o n g them is found by O.Yu. T s v e l o d u b is equal to 1.2161.
as was
We illustrate the m e t h o d c o n s i d e r i n g the d i s s i p a t i v e s p h e r i c a l l y symm e t r i c L a n g m u i r soliton. In many e x p e r i m e n t s on i n t e r a c t i o n of monochomatic e.m.wave with plasma it was observed that in plasma the cavitations of d e n s i t y (cavitons) appear with c h a r a c t e r i s t i c size of order of few Debye lengths. The lifetime of cavitons was much larger than c h a r a c t e r i s t i c time of collapse. The spectral analysis of e . m . w a v e s scattered in these e x p e r i m e n t s show that very n a r r o w e n e r g e t i c spectral lines were located at f r e q u e n c i e s near 2~ and 3/2~ where is f r e q u e n c y of incident wave. These facts can be e x p l a i n e d by the creation of p a r a m e t r i c a l l y fed d i s s i p a t i v e solitons which o r i g i n a t e at places where feeding is e f f e c t i v e n a m e l y where local plasma frequency is equal to ~ or ~/2. The i n t e r a c t i o n of incident wave with these cavitons g e n e r a t e s scattered r a d i a t i o n of waves with combination f r e q u e n c i e s 2~, 3/2~. The linear part of s i m p l i f i e d e q u a t i o n of L a n g m u i r waves c o r r e s p o n d s to linear d i s p e r s i o n equation:
oO=Wp (
~
~
K~÷g
-ZA
r i s i n c r i m e n t , /~ is L a n d a u damping. If we take the n o n l i n e a r term as in Z a k h a r o v
K)
(3)
eq. 7 then we obtain:
R z-~a
8E +.&RE_2R~E+IEI~E 2 + z r E - 6ze
i a7
R , 03
time
t is in units 6 0 ; ¢, ~ is in Debye length u n i t s , E ,p is p l a s m a p r e s s u r e . L a n d a u damping we take is i n c r e m e n t due to p a r a m e t r i c feeding,
(~)
in units in model
form
In contrast w i t h s t a t i o n a r y t u r b u l e n c e we have only one f r e q u e n c y d i s s i p a t i v e s o l i t o n , a n d this helps with l o c a l i s a t i o n in space,In s t a t i o n a r y case we have: a (~)
6.g...~
, Ig=I 1%< 1,
n
in
<< ¢
f here is complex funct£on,/~, is real. Complete f r e q u e n c y in the caviton is~)p(~-~C4).Equation (4) is v a l i d w h e n i n e q u a l i t i s in (5) are satisfied. S u b s t i t u t i n g (5) in (4) we obtain o r d i n a r y differential eq.with ~ o u n d a r y c o n d i t i o n s f ( 0 ) ~ f(oo)= O. _O. is e i g e n v a l u e d e p e n d i n g on r .This e q u a t i o n with help of l i n e a r Green's function takes the form of n o n l i n e a r integral equation.
V.I. Petviashvili /Multidimensional and dissipative solitons
332
o
(6)
~']~'z
-,
-,
G~ = ~ + } K % 2 K C
,/a
ZF~ +
~SZk'°
(8)
- is B e s s e ! f u n c t i o n , O~- is F u r i e r - B e s s e l s p e c t r u m of l i n e a r oper a t o r in ( h ) . E q . ( 6 ) has s o l i t o n s o l u t i o n as far as i n t e g r a l (7) is regular. I t e r a t i o n m e t h o d a p p l a i e d to (6) gives d i v e r g i n g s e q u e n c e of aprox i m a t i o n s . T o m a k e the s e q u e n c e c o n v e r g i n g we i n t r o d u c e the s t a b . f a c tor s: ~ ~
o
When
f is s o l u t i o n
@
of
(6) we
have:
s = l. So we
instead
of
(6)
solve
R i g h t h a n d side of (i0) - has d e g r e e of n o n l i n e a r i t i e q u a l to O. W h e n we solve (i0) by i t e r a t i o n s d i v e r g i n g e n c e d i s a p e a r s but the se~ q u e n c e is not c o n v e r g i n g u n t i l we find c o r r e c t e i g e n v a l u e ~ l .When we s e l e c t p r o p e r e i ~ e u v a l u e the s e r i e s b e c o m e q u i c k l y c o n v e r g i n g , and at the same time s tends to i. In f o l l o w i n g t a b l e the e i g e n v a l u e s . ~ , are given efficients in s p e c t r u m of i n c r e m e n t /~ in cases 2 .lO -4
1.5.1o -3
corresponding whenC-a(#l~Km
to co2K#I#~l
0.oo3
o,o0~
0,Ol
o,o16
0,23
~(i~ 1 . 4 . 1 0 -3 7.10 -3
0,01
0,014
0,028
O,Oh
O,O56
~3
0,02
0,06
0,08
0,I
0,16
0,2
/~-
0,027
O,O57
0,08
0 I
0,14
0,17
~
~,o,..-
~
js9
Q ego Figure
~ol
0 f
I
,
gd
2.
D e p e n d e n c e of e l e c t r i c f i e l d E i n ~ - - ~ u n i t s s o l i t o n centre in D e b y e radius u n i t s . Figure
0,2
3.
Deoendence of maximal dencity of energy ing distance from s o l i t o n c e n t r e -~ _ and on r e l a t m v e f r e q u e n c y of solmton.~..
on
!
distance
from
= , E ~! of corresponds o l i t o n e n e r g y $ -- ~?TSIE%Itdz
V.[. Petviashvili / Multidimensional and dissipative solitons
333
O t h e r r e s u l t s are g i v e n on f i g u r e s 2 , a n d 3 . L i k e s o l i t o n s on v i s c o u s l i ~ u l d film c a v i t o n s are r a n d o m l i d i s t r i b u t e d in u n s t a b l e p l a s m a . L i f e t i m e of c a v i t o n s is d e t e r m i n e d by c o l l i s i o n s w i t h n e i g h b o u r e s . C a v i t o n t u r b u l e n c e is c h a r a c t e r i s e d by d i s c r e t e s p e c t r a l lines of f r e q u e n c l e s , w h i l e t u r b u l e n t or c o l l a p s i n g L a n g m u i r o s c i l l a t i o n s p o s ses w i d e f r e q u e n c y s p e c t r u m . In the s t e l l a r a t o r s in r e g i m e s w h e n ~ p < ~ ¢ a n d w i t h l o n g i t u d i n a l e l e c t ! ric f i e l d less than D r e i s e r s p o w e r f u l l l a n g m u i r o s c i l l a t i o n s l o c a l i s e d on s m a l l r a d i u s and on p o l o i d a l a n g l e w e r e o b s e r v e d . The l o c a l i s a t i o n r a n g e c o i n c i d e d w i t h r u n a w a y e l e c t r o n s w h i c h i n d i c a t e s that l a s t ones w e r e f e e d i n g L a n g m u i r w a v e s o l i t o n s a p p a r e n t l y by the anomalous Dopier effect . The s p e c t r u m of e x c i t e d w a v e s was very n a r r o w . The h i g h d e g r e e of m o n o c h r o m a t i c i t y and l o c a l i s a t i o n of this w a v e s can not be e x p l a i n e d by the w e a k t u r b u l e n c e t h e o r y . We s h o w that in this case the e x i s t e n c e of L a n g m u i r w a v e s o l i t o n p e r i o dical a l o n g m a g n e t i c f i e l d and l o c a l i s e d a c r o s s the one is p o s s i b l e . The d e n s i t y w e l l can not a r i s e in the p r e s e n c e of r u n a w a y e l e c t r o n s ( t a i l e on the e l e c t r o n d i s t r i b u t i o n f u n c t i o n ) b e c a u s e the w a v e s p r o p a g a t i n g r e v e r s e l y to the r u n a w a y e l e c t r o n s are d u m p e d by the n o r m a l D o p p l e r e f f e c t . S o o n l y n o n l i n e a r i t i e s of w a v e s p r o p a g a t i o n g in tale d i r e c t i o n s t a y acting. T h e n e q . o f s o l i t o n in w h i c h d e p e n d e n c e on coo r d i n a t e alon 9 m a g n e t i c f i e l d z and t i m e t o c c u r o n l y in c o m b i n a t i o n ~-~p U ~ ~ t a k e s the form: ~ 2e~
Here ~ i s
electric
potential,angular
brackets
!fl~|~< 4 , a f t e r e x p a n d i n g (ii) in p o w e r s o l u t i o n in the form of p l a n e s o l i t o n :
denote
series
of~
averaging we
by Z.
easily
find
~ h e n r u n a w a y e l e c t r o n s p u m p the s o l i t o n its a m p l i t u d e A g r o w s and a d i t i o n a l h a r m o n i c s a r i s e t i l l the h a r m o n i c w i t h w a v e l e n g t h of order of D e b y e l e n g t h g r o w s s u f f i c i e n t l y large. Th~ soliton becomes dissipative with amplitude~%and eq.(I) is to be s o l v e d by the m e t h o d of stab factor. The a x i a l l y s i m m e t r i c s o l u t i on can be o b t a i n e d . D r i f t w a v e s also f o r m a x i a l l y s y m m e t r i c s o l i t o n s but they are i n c l i n e d by small a n g l e to d i r e c t i o n of m a g n e t i c f i e l d and p r o p a g a t e w i t h v e l o c i t y m o r e than e l e c t r i c d r i f t one. The m o s t i m p o r t a n t n o n l i n e a r i t y for d r i f t w a v e s s o l u t i o n is c a u s e d by e l e c t r o n t e m p e r a t u r e g r a d i e n t . I o n i c or e l e c t r o n i c c i c l o t r o n w a v e p a c k e t s g e n e r a t e d i a m a g n e t i c c u r r e n t due to hf p r e s s u r e g r a d i e n t across the m a g n e t i c f i e l d , w h i c h c r e a t e s m a g n e t i c w e l l in sites of w a v e p a c k e t s [ 9 1 The f r e q u e n c y of c i c l o t r o n w a v e s is as s e n s i t i v e to m a g n e t i c f i e l d v a r i a t i o n s as L a n g m u e r w a v e f r e q u e n c y to d e n c i t y v a r i a t i o n s . S o like L a n g m u i e r w a v e s c i c l o t r o n w a v e s e l f l o c a l i s a t i o n is p o s s i b l e . F o u n d the r e l a t i o n b e t w e e n w a v e e n e r g y d e n c i t y and m a g n e t i n g w e l l a m p l i t u d e it is e a s y to c a l c u l a t e p a r a m e t e r s of c i c l o t r o n s o l i t o n s [~.
REFERENCES i. P e t v i a s h v i l i V . l . , F i s i k a P l a s m i , 2 ( 1 9 7 6 ) 469 2. K a d o m t s e v B . B . , P e t v i a s h v i l i V . I . D o k l a d i Akad. N a u k 753 3. P e t v i a s h v i l i V.I. F i s i k a P l a s m i , l (1975) 28
SSSR,192
(1970)
334
V.L Petviashvili / Multidimensional and dissipative solitons
h. Manakov S.V.,Zakharov V.E.,Bordag L.A.,Its A.R.,Matveev V.B., Phisics Lett.,63A (1977) 205 5. Petvlashvili V.I.,Tsvelodub O.Yu.,Dokladi Aead. Nauk SSSR,238 (1978) 1321 6. Benjamin T.B.,Fluld. Mech.,2 (1957) 55h 7. Zakharov V.E.,JETP,62 (1972) 17h5 8. Petvlshvili V.I., Fisika Plasmi,3 (1977) 270 9. Nekrasov A.K.,Petviashvi!i V.I.,JETP,77 (1979) 605.