Stable spatial envelope wave structures in plasmas
Conditions for formation of two- and three-dimensional localized envelope soliton-like structures in plasmas are investigated. These structures are described by modified nonlinear Schrцdinger equation, including higher order linear and nonlinear effects. It is shown that higher order effects are cru...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-784172015-03-17T03:02:19Z Stable spatial envelope wave structures in plasmas Davydova, T.A. Yakimenko, A.I. Zaliznyak, Yu.A. Magnetic confinement Conditions for formation of two- and three-dimensional localized envelope soliton-like structures in plasmas are investigated. These structures are described by modified nonlinear Schrцdinger equation, including higher order linear and nonlinear effects. It is shown that higher order effects are crucial to explain formation of localized structures, which have been observed in plasmas. Stability of stationary soliton-like structures is investigated. Obtained results are applied to interpretation of the experiments on Langmuir solitons generation in electron-beam plasma systems. Вивчено умови формування двовимірних та тривимірних солітоноподібних структур огинаючих в плазмі. Такі структури описуються модифікованим нелінійним рівнянням Шредінгера, що враховує лінійні ті нелінійні ефекти вищого порядку. Показано, що ефекти вищого порядку є принципово важливими для пояснення утворення локалізованих квазідвовимірних та тривимірних структур огинаючих, що спостерігаються в плазмі. Досліджено стійкість стаціонарних солі тонних структур. Отримані результати використовуються для пояснення експериментів із збудження ленгмюрових солітонів у плазмово-пучкових системах. Изучены условия формирования двумерных и трёхмерных солитоноподобных структур огибающих в плазме. Такие структуры описываются модифицированным нелинейным уравнением Шрёдингера, учитывающим линейные и нелинейные эффекты высшего порядка. Показано, что эффекты высшего порядка принципиально важны для объяснения образования локализованных сруктур огибающих, которые наблюдаются в плазме. Исследована устойчивость стационарных солитонных структур. Полученные результаты применяются для интерпретации экспериментов по возбуждению ленгмюровских солитонов в плазменно-пучковых системах. 2005 Article Stable spatial envelope wave structures in plasmas / T.A. Davydova, A.I. Yakimenko, Yu.A. Zaliznyak // Вопросы атомной науки и техники. — 2005. — № 1. — С. 13-17. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 52.35.Sb, 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/78417 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Davydova, T.A. Yakimenko, A.I. Zaliznyak, Yu.A. Stable spatial envelope wave structures in plasmas Вопросы атомной науки и техники |
| description |
Conditions for formation of two- and three-dimensional localized envelope soliton-like structures in plasmas are investigated. These structures are described by modified nonlinear Schrцdinger equation, including higher order linear and nonlinear effects. It is shown that higher order effects are crucial to explain formation of localized structures, which have been observed in plasmas. Stability of stationary soliton-like structures is investigated. Obtained results are applied to interpretation of the experiments on Langmuir solitons generation in electron-beam plasma systems. |
| format |
Article |
| author |
Davydova, T.A. Yakimenko, A.I. Zaliznyak, Yu.A. |
| author_facet |
Davydova, T.A. Yakimenko, A.I. Zaliznyak, Yu.A. |
| author_sort |
Davydova, T.A. |
| title |
Stable spatial envelope wave structures in plasmas |
| title_short |
Stable spatial envelope wave structures in plasmas |
| title_full |
Stable spatial envelope wave structures in plasmas |
| title_fullStr |
Stable spatial envelope wave structures in plasmas |
| title_full_unstemmed |
Stable spatial envelope wave structures in plasmas |
| title_sort |
stable spatial envelope wave structures in plasmas |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2005 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78417 |
| citation_txt |
Stable spatial envelope wave structures in plasmas / T.A. Davydova, A.I. Yakimenko, Yu.A. Zaliznyak // Вопросы атомной науки и техники. — 2005. — № 1. — С. 13-17. — Бібліогр.: 20 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT davydovata stablespatialenvelopewavestructuresinplasmas AT yakimenkoai stablespatialenvelopewavestructuresinplasmas AT zaliznyakyua stablespatialenvelopewavestructuresinplasmas |
| first_indexed |
2025-07-06T02:31:21Z |
| last_indexed |
2025-07-06T02:31:21Z |
| _version_ |
1836863016748974080 |
| fulltext |
STABLE SPATIAL ENVELOPE WAVE STRUCTURES IN PLASMAS
T.A. Davydova, A.I. Yakimenko, and Yu.A. Zaliznyak
Plasma Theory Dept., Institute for Nuclear Research, Kiev 03680, Ukraine
Conditions for formation of two- and three-dimensional localized envelope soliton-like structures in plasmas are
investigated. These structures are described by modified nonlinear Schrцdinger equation, including higher order linear
and nonlinear effects. It is shown that higher order effects are crucial to explain formation of localized structures, which
have been observed in plasmas. Stability of stationary soliton-like structures is investigated. Obtained results are applied
to interpretation of the experiments on Langmuir solitons generation in electron-beam plasma systems.
PACS: 52.35.Sb, 52.35.Mw
INTRODUCTION
Formation of envelope coherent wave structures in plasmas
and other nonlinear dispersive media can be rather
universally described by nonlinear Schrцdinger equation
(NSE) with some additional terms. Nonlinear wave structure
in the framework of NSE is often a subject to collapse – its
size decreases and intensity grows, forming singularity in a
finite time. However, for intense and narrow wave packets,
the higher-order effects are to be taken into account to
describe packet’s evolution correctly. In the situation, when
dissipation is not very essential, those higher-order effects are
usually associated with the saturation of local nonlinearity,
nonlinear dispersion (nonlocality) and with high-order wave
dispersion. Spatial envelope wave structures with 2D
azimuthal or 3D spherical symmetry may be described by the
generalized NSE (GNSE) of the form
i ∂ψ
∂ t
DΔr ψPΔr
2ψBψ∣ψ∣2
Kψ∣ψ∣4CψΔr∣ψ∣
2=0
, (1)
where Δr is the radial part of the Laplacian, terms
proportional to D and P describe second and fourth order
dispersion effects, terms proportional to B and K represent
saturable cubic-quintic nonlinearity (BK<0), term
proportional to C describes nonlocal interaction of the main
wave with other waves or it’s self-interaction.
In different physical situations the importance of these
higher-order terms may be different. For instance, formation
of quasi-one-dimensional upper-hybrid solitons which have
been observed experimentally in [1], can not be described in
the framework of the model of GNSE (1) with B=K=P=0,
DC<0 proposed in [2], since these model solitons were
shown to be unstable with respect to collapse in [3]. At the
same time, in [4,5] fourth-order linear dispersion was
demonstrated to stabilize upper-hybrid solitons in the
conditions of experiment [1]. The model, developed in [2]
appears again in the theoretical attempts to explain formation
of radially-elongated structures – streamers in tokamaks and
their influence on the anomalous transport processes [6]. As
it is well-known, in the space of more than one dimension,
NSE (1) with C=K=P=0 has localized solution which is
unstable with respect to collapse if DB>0 and has no
localized solutions at all, if DB<0. In the first case (DB>0)
any of higher-order effects may be sufficient to prevent
collapse and stabilize soliton solutions of Eq. (1). The
stabilizing effect of saturable cubic-quintic
nonlinearity [BK<0 in (1)] has been thoroughly
investigated in the context of nonlinear optics both in
2D and 3D cases [7]. Effects of nonlocal nonlinearity
(with BC>0) in 2D space were first studied in
application to upper-hybrid solitons in plasmas in [8]
and to formation of matter waves in Bose-Einstein
condensates [9]. Simultaneous influence of high-order
dispersion and cubic-quintic nonlinearity on the
formation of whistler wave stationary waveguides in
normal and anomalous dispersive regimes has been
studied in [10]. In the case DB<0, the proper
combination of higher order linear and nonlinear
effects is able to explain formation of localized
coherent structures observed in many experiments.
Taking into account higher-order linear dispersion
together with nonlinear dispersion (such that DP>0,
DB<0, BC>0 in (1)), the observations [11] of strongly
localized upper-hybrid structures in the anomalous
dispersion regime were explained in [12]. A new type
of stable soliton solutions, the so-called chirped
solitons with nonlinearly changing phase were found
analytically and numerically in 1D and in 2D [13-15].
In this paper we consider 2D and 3D soliton
solutions of Eq. (1) with BD>0 in the presence of
effects which stabilize the collapse. In Section 2 we
show that in the 2D space stabilizing effects from all
additional terms (proportional to K, C, and P) are
summarized in a simple way. We show that in the
presence of quintic (BK<0) nonlinearity only the wave
packet in the diffractionless case [16] (DH<0, H being
the Hamiltonian), can not be contracted to the
infinitely small size for any value of wave power N
and in the space of any dimension (d=1,2,3). In
Section 3 we investigate analytically and numerically
the stabilizing effect of nonlocal nonlinearity on 3D
Langmuir wave structures in plasmas.
COLLAPSE ARREST MECHANISMS
The equation (1) has the integrals: number of
quanta
N=∫∣ψ∣2 d r ,
Hamiltonian
Problems of Atomic Science and Technology. 2005. № 1. Series: Plasma Physics (10). P.13-17 13
H=D∫∣∇ψ∣2 d r−P∫∣Δ¿ψ∣
2 d r− B
2∫∣ψ∣4 d r−
∣∇ψ∣2 d r≡¿
−
K
3 ∫∣ψ∣6 d rC
2 ∫ ¿≡DI D−PI P−
B
2 I B−
K
3 I K
C
2 I C .
Also we choose in this paper the following signs of
coefficients: D>0, B>0, P<0, K<0, C>0. In this case it is easy
to show (similarly to that was done in [16]), that if the
Hamiltonian is negative, any wave packet can not decrease
it’s amplitude |ψmax| infinitely. Indeed,
∣H∣ B
2∫∣ψ∣4 d r B
2
N ∣ψ max∣
2 ,
and so that
∣ψ max∣
2
2 ∣H∣
BN
.
Therefore, for the negative Hamiltonian (H<0) wave packet
will not disperse. Using the Gцlder inequality
I B
2 I K N , (2)
one estimates
DI D
B
2
I B−
K
3
I K
B
2
I B−
K
3
I B
2
N
3
16
B2 N
K
.
(3)
Further, employing the “uncertainty relation”
1
N ∫∣ψ∣2r 2 d r≡r eff
2 d
4
N
I D
(4)
where d is the number of space dimensions, we get
r eff
2 d
4
N
I D
d
4
16
3
KD
B2 =4d
3
KD
B2 . (5)
It is interesting to note that the minimum possible square
radius of any wave packet in the case K≠0 does not depend
on N in space of any dimension, but it increases with number
of dimensions. Equation (5) gives also the estimate of
minimum square radius for soliton solution if H<0. Note that
this statement is true also for saturable nonlinearity which is
typical for plasma waves, when instead of
B∣ψ∣2 ψ−K∣ψ∣4 ψ in Eq. (1) we have
Bψf ∣ψ∣2=Bψ
1 −exp −κ∣ψ∣2
κ
.
Then we get estimate for the Hamiltonian
H≥DI D−B∫[∫0
∣ψ∣2
f x dx]d r=
DI D−
B
κ2 ∫ [κ∣ψ∣2−1 −exp−κ∣ψ∣2]d rDI D−
BN
κ
If H<0 then DI DBN / κ and thus r eff
2 d Dκ / 4B .
Let us consider in more details 2D space, which corresponds,
in particular, to stationary propagation of wave beams. In this
case, from the integral inequality
I D≥5 . 84 I B /N (6)
we see that if H<0 then
B
2
I BDI D≥D
5 . 84 I B
N
(7)
or N>21.36 D/B=Ncr. In the other hand, in 2D space
and for P=0, the virial relation takes rather simple
form
N
d 2 reff
2
dt 2 =8D[H
∣K∣
3
I K
C
2
I C] .
(8)
For soliton solutions the r.h.s. of Eq. (8) is equal to
zero, which means that H<0 if any of the coefficients
(K or C) does not vanish. Thus, the condition (7) is a
necessary condition for soliton formation in 2D space.
We have seen before that if K≠0, H<0, the wave
packet can non contract infinitely.
Suppose that N<Ncr. In this case H>0 and the
relation (8) can be rewritten as
N
d 2 reff
2
dt2 =8D[2H−DI D
B
2
I B]¿
¿8D[2H−DI D
BN
11.7
I D]
,
where we have used the integral inequality (7). Using
also the relation (8), which gives
N
d 2 reff
2
dt 2 8 DH , we get
8 DHN
d 2 r eff
2
dt2 8D
N [2H−D1 − N
N cr I D]
.
For N<Ncr :
8 DH8D[2H− 1
r eff
2 D1 − N
N cr N ] , or
r eff
2 DN
2H 1 − N
N cr . (9)
Thus, for N<Ncr, the wave packet collapse is also
impossible.
Let us show that in the 2D space, Hamiltonian is
bounded from below if D≥0, B>0, C≥0, P≥0, K≤0, and
at least one of the coefficients C, P or K does not
vanish. In addition to (7), (8), integrals IC and IP are to
be estimated in terms of integral IB. Using Gölder
inequalities we have
∫∣ψ∣8d r≥ 1
N ∫∣ψ∣5 d r 2≥ I B
4
N∫∣ψ∣3d r
≥
I B
3
N 2
.
With the help of inequality (7) this gives the
estimation
I C≥
5 . 84
I B
∫∣ψ∣8 d r≥5 . 84
I B
N 2 . (10)
Also we have
I P≥
1
N ∫∣∇ ψ∣2d r 2≥5 . 842
I B
2
N 3
.
(11)
Thus
H≥D
5 . 84 I B
N
− B
2
I B
∣K∣
3
I B
2
N
C
2
5 . 84 I B
2
N 2
5 .842∣P∣ I B
2
N 3 ≥
¿−1
16
BN−11 .7D2
KN /35 . 84С5 . 842∣P∣/N
which was required to prove. For any soliton solution, H<0.
Thus, there exists at least one soliton state which gives
minimum to the Hamiltonian and hence is stable.
STABLE LANGMUIR SOLITONS
Here the impact of nonlocal nonlinearity C>0 will be studied
here in the connection with the Langmuir solitons (LS).
Formation of LS are connected with the plasma extrusion
from the regions of strong high-frequency electric field and
trapping of plasmons into the formed density well (caviton).
However, in the previous simplified theoretical models, 2D
and 3D solitons occur to be unstable with respect to a
collapse: above some threshold power. Nevertheless,
experimental observations [17,18] of 3D Langmuir collapse
demonstrate saturation of wave-packet's spatial scale at some
minimum value being of order few tens of Debye radii. It
have been observed in Refs. [17,18] that at times t > 50ωPi,
where ωPi is the ion plasma frequency, Langmuir wave
packets show considerably slow dynamics (subsonic regime).
To our best knowledge, these observations do not meet an
appropriate theoretical explanation yet. As it is shown in
[19], the local part of electron-electron nonlinearity
counteracts the contraction of wave packet. At the same time,
the nonlocal contribution of the additional nonlinear term was
omitted in [19], though it is of great importance for
sufficiently narrow and intense wave packets. As it will be
shown below, the role of nonlocal nonlinearity is
quantitatively even more significant.
We consider subsonic motions and neglect the terms
with time derivatives in the second equation of above set. As
result, this set is reduced to the single partial differential
equation:
i ∂E
∂ t
D∂
∂ r
r−2 ∂
∂r
r2 EBE∣E∣2
CE Δr∣E∣
2−Γ E∣E∣2
r2 =0
(12)
where the coefficients D, B, C, Г are given by the
expressions:
D= 3
2
ω p r D
2 , B=
ω p
32 π Mn0 cs
2 ,
C= 7
96 π mn0 ω p
, Γ= 1
48π mn0ω p
.
The nonlinear part of this equation includes common cubic
nonlinearity (term proportional to B) as well as nonlocal
(term proportional to C) and local (term with Γ) parts
of electron-electron nonlinearity. Let us show that the
effective width reff (4) of any stationary and non-
stationary wave packet governed by Eq. (12) is
bounded from below in the most interesting case of
self-trapped wave packets having negative
Hamiltonian
H=D∫∣r−2∂r r 2 E∣2 d r−0 . 5 B∫∣E∣4 d r
0 . 5 C∫ ∇∣E∣22 d r0 .5 Γ∫ r−2∣E∣4 d r0
Using the inequality
∫∣∇ E∣2d r≥0 . 25∫ r−2∣E∣2d r ,
which is valid in 3D space, one finds for H<0:
B∫∣E∣4d rC∫ ∇∣E∣2 2 d rΓ∫ r−2∣E∣4 d r≥
¿ ΓC /4∫ r−2∣E∣4 d r
. (13)
On the other hand we have
∫ αr∣E∣−r−1∣E∣2d r=α 2∫ r 2∣E∣2 d r∫ r−2∣E∣4 d r−
−2α∫∣E∣3 d r≥0
for any positive α. Using also Gцlder inequality
∫∣E∣3 d r≤∫∣E∣2 d r 1/2∫∣E∣4 d r 1/2 ,
we obtain for any α
α2∫ r 2∣E∣2 d r−2α ∫∣E∣2 d r 1/2∫∣E∣4 d r 1/2
∫ r−2∣E∣4d r≥0
.
Requirement that the discriminant of the l. h. s. of the
last inequality is negative gives
r eff
2 ≥∫∣E∣4 d r /∫ r−2∣E∣4 d r .
Finally, using also the relation (13) we get
r eff
2 ΓC / 4
B .
Soliton solutions have a form
E=ψ r exp−i λτ , where λ is the nonlinear
frequency shift of the soliton. To obtain qualitative
information about the soliton properties let us consider
the variational approach with trial function
E r =N 1/2 a−3/2 r /a exp −r 2 /a2=
¿N 1/2 f ξ
,
(14)
so that ∫∣E∣2 d r=N . Substitution of (14) into the
Hamiltonian gives
Problems of Atomic Science and Technology. 2005. № 1. Series: Plasma Physics (10). P.13-17 15
H
N
=
DI d
a2 −
BI b N
a3
CI cΓI γN
a5
,
where
I d=4π∫0
∞ [ξ−2∂ξ ξ 2 f ]2ξ 2 d ξ ,
I b=2π∫0
∞
f 4ξ 2 dξ , I c=2π∫0
∞ ∂ ξ f 22 ξ 2 dξ ,
I γ=2π∫0
∞
f 4dξ .
The standard variational procedure gives connection between
number of quanta N an characteristic size of the soliton a:
N= 2 da3
3 ba2−5c
, (15)
where d=DId, b=BIb, c=CIc+ΓIγ.The dependence N(a) which
follows from the Eq. (15) is similar to that presented in Fig.
1. It follows from (15) that a2>5c/(3b)=a2
min. In the other
hand, N>Nmin=(d/b)aex, where aex=(5c/b)1/2>amin. When soliton
solution is perturbed, tha phase variation with spatial
coordinates appears. Taking this into account, one can find
for small deviationa of parameter a from soliton’s parameter
a0 which satisfies the equation (15) (see, e.g. [20]):
N
2d
d 2
dt 2 a−a0
∂2 H
∂ a2 a−a0=0 .
0.00 0.05 0.10 0.15 0.20
1100
1200
1300
1400
1500
1600
1700
N
um
be
r
of
q
ua
nt
a N
Nonlinear frequency shift λ
Fig. 1. The number of quanta versus the nonlinear frequency
shift for 3D Langmuir solitons. Variational prediction is
plotted by the dashed line
1200 1400 1600 1800
6
8
10
12
14
16
18
0 10 20 30 40
0.0
0.2
0.4
0.6
0.8
1.0
ψ
R
0.15
0.04
0.015
R
ef
f
N
Fig. 2. The effective soliton radius versus the number
of quanta. Dashed line presents variational
dependence. On the inset, the soliton profiles are
presented for different λ
Soliton is stable if ∂2 H /∂ a20 . It is easy to
show that solitons are stable if they belong to the
lower branch of the curve N(a0) (15), when N<Nmin
and amin<a0
2<aex. From our variational approach it
follows that
∂N
∂ a
= ∂2 H /∂ a2
−∂2 H /∂a ∂N
=∂2 H /∂ a2
∂ λ /∂ a
,
and hence
∂N
∂ λ
=∂N
∂ a
∂ a
∂ λ
=∂2 H
∂ a2 ∂a
∂ λ
2
.
Thus, in the framework of the variational approach,
the soliton stability condition ∂2 H /∂ a20
coincides with Vakhitov-Kolokolov criterion.
Variational results were found to be in a very good
agreement with our numerical calculations.
The radial soliton profiles ψ(R) are found from the
equation:
−λψ∂2ψ
∂ R2
2
R
∂ψ
∂ R
−2
R2 ψψ∣ψ∣2
7
2
ψΔR∣ψ∣
2−
ψ∣ψ∣2
R2 =0
,
were the dimensionless variables are used:
R= 3
2
r
r D
, τ= 9
4 ω p t , ψ=E /72 πn0 T e .
Stationary states of Eq. (12) were investigated
numerically, and the results are summarized in Figures
1 and 2. Figure 1 presents the energy dispersion
diagram, number of quanta versus the nonlinear
frequency shift for 3D Langmuis solitons, while in the
Fig. 2, the effective soliton’s radius defined by (4) is
plotted versus the soliton power. In 3D, there are two
soliton branches, one , corresponding to
∂ N /∂ λ0 , is stable, and the other is unstable. It
is clear, that analytical predictions found by the
approximate variational approach (dashed lines in
figures) are in a good agreement with our exact
numerical calculations. With the increase of soliton
power, the effective radius decreases and saturates at
values being of order 6rDe.
CONCLUSIONS
Saturation of the nonlinearity, as well as non-local
wave interaction is able to arrest the wave collapse.
The new estimations for minimum possible size of any
stationary and non-stationary wave packets are found.
Stable solitons may be formed due to any of these
higher order nonlinear effects. We have performed
analytical and numerical studies of spatial Langmuir
solitons in the framework of model based on
generalized nonlinear Schцdinger equation including both
local and nonlocal electron-electron nonlinearities. Their
influence on intense and narrow Langmuir wave packets are
of the same order. Any of them is able to arrest the Langmuir
collapse. Both nonlinearities lead to the saturation of soliton
width with an increase of the energy, but quantitatively the
effect of nonlocal nonlinearity is more significant. In the 3D
case, two soliton branches coexist, one is stable and the other
is unstable.
REFERENCES
1. T. Cho, S. Tanaka Observation of an upper-hybrid
soliton // Phys. Rev. Lett. (45). 1980, p. 1403.
2. M.V. Porkolab, M.V. Goldman Upper-Hybrid Solitons
and Oscillating two-stream instabilities // Phys. Fluids
(19). 1976, p. 872.
3. A.G. Litvak, A.M. Sergeev.On one-dimensional collapse
of plasma waves // Pis’ma ZhETF (21). 1978, p.549.
4. T.A. Davydova, A.I. Fishchuk. Bistable Upper-Hybrid
Solitons // Phys. Scr. . (57). 1998, p. 118.
5. T.A. Davydova, A.I. Fishchuk. Two Kinds of Upper-
Hybrid Soliton Bistability // Phys. Lett. 1998. v.A245.
p.453.
6. Ц.D. Gьrcan, P.H. Diamond. Streamer formation and
collapse in electron temperature gradient driven
turbulence // Phys. Plasmas (11). 2004, p. 572.
7. A. Desyatnikov, A. Maimistov, B. Malomed. Three-
dimensional spinning solitons in dispersive media with
cubic-quintic nonlinearity // Phys. Rev. E. (61). 2000, p.
3107.
8. T.A. Davydova, A.I. Fischuk. Upper hybrid nonlinear
wave structures // Ukr. Journ Phys. (40). 1995, P.487.
9. W. Krolikowski et. al. Modulational instability, solitons
and beam propagation in spatially nonlocal nonlinear
media // J. Opt. B. (6). 2004, p. 5288.
10. T.A. Davydova, A.I. Yakimenko, Yu.A. Zaliznyak.
Two-dimensional solitons and vortices in normal and
anomalous dispersive media // Phys. Rev. E. (67) 2003,
P.026402.
11. P.J. Christiansen, V.K. Jain, L. Stenflo. Filamentary
collapse in electron-beam plasmas // Phys. Rev. Lett.
(18). 1981, p. 1333.
12. T.A. Davydova and A.I. Yakimenko. Spatial solitons in
anomalous dispersive media with nonlocal nonlinearity //
Ukr.Journ. of Phys. (18). 2003, p. 623.
13. T.A. Davydova, Yu.A. Zaliznyak. Chirped solitons near
plasma resonances in magnetized plasmas // Phys. Scr.
(61). 2000, p. 476.
14. T.A. Davydova, A.I. Yakimenko. Whistler waves
self-focusing in laboratiory and ionospheric
plasma in density troughs // VANT (4). 2003,
p. 119.
15. T.A. Davydova, Yu.A. Zaliznyak. Schrцdinger
ordinary solitons and chirped solitons: fourth-
order dispersion effects and cubic-quintic
nonlinearity // Physica D. (156). 2001, p. 260.
16. V.E. Zakharov, V.E. Sobolev, and V.C. Synakh.
Light beam propagation in nonlinear media //
Sov. Phys. JETP (33). 1971, p.77.
17. A.Y. Wong, P.Y. Cheung. Three-dimensional
collapse of Langmuir wave // Phys. Rev. Lett
(52). 1984, p.1222.
18. P.Y. Cheung, A.Y. Wong. Nonlinear evolution of
electron beam-plasma interactions // Phys. Fluids
B. (28). 1985, p. 1538.
19. E.A. Kuznetsov. Average description of plasma
wave // Sov. J. Plasma Phys (2). 1976, p. 187.
20. T.A. Davydova, A.I. Yakimenko. Stable
multicharged localized optical vortices in cubic-
quintic nonlinear media // J.Opt A. 2004. A. V.6.
S197.
УСТОЙЧИВЫЕ ПРОСТРАНСТВЕННЫЕ ВОЛНОВЫЕ СТРУКТУРЫ ОГИБАЮЩИХ
В ПЛАЗМЕ
Т.А. Давыдова, А.И. Якименко, Ю.А. Зализняк
Изучены условия формирования двумерных и трёхмерных солитоноподобных структур огибающих в плазме.
Такие структуры описываются модифицированным нелинейным уравнением Шрёдингера, учитывающим
линейные и нелинейные эффекты высшего порядка. Показано, что эффекты высшего порядка принципиально
важны для объяснения образования локализованных сруктур огибающих, которые наблюдаются в плазме.
Исследована устойчивость стационарных солитонных структур. Полученные результаты применяются для
интерпретации экспериментов по возбуждению ленгмюровских солитонов в плазменно-пучковых системах.
Problems of Atomic Science and Technology. 2005. № 1. Series: Plasma Physics (10). P.13-17 17
СТІЙКІ ПРОСТОРОВІ ХВИЛЬОВІ СТРУКТУРИ ОГИНАЮЧИХ В ПЛАЗМІ
Т.О. Давидова, О.І. Якименко, Ю.О. Залізняк
Вивчено умови формування двовимірних та тривимірних солітоноподібних структур огинаючих в плазмі. Такі
структури описуються модифікованим нелінійним рівнянням Шредінгера, що враховує лінійні ті нелінійні
ефекти вищого порядку. Показано, що ефекти вищого порядку є принципово важливими для пояснення
утворення локалізованих квазідвовимірних та тривимірних структур огинаючих, що спостерігаються в плазмі.
Досліджено стійкість стаціонарних солі тонних структур. Отримані результати використовуються для
пояснення експериментів із збудження ленгмюрових солітонів у плазмово-пучкових системах.
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