Stable multisolitary structures in plasmas with nonlocal nonlinearities

A.I. Yakimenko¤,†, V.M. Lashkin¤ and O.O. Prikhodko†
¤Institute for Nuclear Research NAS of Ukraine, 47 Prospect Nauki, 03680 Kiev, Ukraine
†Physical Department, Taras Shevchenko National University, Prospect Glushkova 2, Kiev 03680, Ukraine
Abstract. We investigate the bound states of out-of-phase two-dimensional bright solitons in plasmas with thermal nonlocal
nonlinearity. The general properties and stability of these multisolitary structures are studied analytically and numerically.We
have found that a steady bound state of localized coherent solitary structures can exist above some threshold power. A dipolar
multisoliton occurs to be stable within the finite range of the beam power. The bound states of three or four solitons appear to
be unstable: they decay into flying off single solitons or merge into one monopole with emission of free waves.
Keywords: Solitons, vortices, nonlinear phenomena in plasmas, nonlinear Schrödinger equation
PACS: 52.35.Mw, 42.65.Tg
A prominent feature of various plasma systems is the spatial nonlocality of the nonlinearity. Nonlocality means that
nonlinear media response depends on the wave packet intensity at some extensive spatial domain, which is typical for
plasmas with thermal diffusion [1, 2] or additional spatial dispersion [3]. An important property of spatially nonlocal
nonlinear response is that it prevents a catastrophic collapse which usually occurs in local self-focusing media when
the power of the two- or three-dimensional wave packet exceeds some critical value. In the absence of collapse in
nonlocal nonlinear media various single and composite solitary structures have been predicted and experimentally
demonstrated. A composite soliton structure, or a multi-soliton complex is a self-localized state which is a nonlinear
superposition of several fundamental solitons. Stationary multi-soliton structures in nonlocal media were considered
first in Refs. [4, 5]. However, the question of the stability of the multisolitons is still an open question. In this report,
using direct 2D simulations, we find a class of radially asymmetric two-dimensional soliton solutions and show that,
at some input power, the dipole-mode solutions are stable while the tripoles and quadrupoles are always unstable.
We consider here the stationary (in time) self-focusing of the intense wave beam in plasmas. The basic system of
equations, written in appropriate dimensionless variables, is

where D? = ¶ 2=¶ x2 +¶ 2=¶ y2 is the transverse Laplacian. Equations (1) and (2) describe the propagation in zdirection
of the electric field envelope Y(x;y; z) coupled to the temperature perturbation q (x;y; z) in a plasma [2].
The dimensionless variables (x;y; z) and (y;q ) are connected with the corresponding physical variables (X;Y;Z) and

where E is the electric field envelope, Q = T0
e =T is the relative electron temperature perturbation, T being unperturbed
temperature, le is the thermodiffusion length, k0 is the wave number, parameter a characterizes the relative energy that
electron delivers to a heavy particle with mass M during single collision (a2 ¼ 2m=M), E2
c = 3mT(w2
0 +n 2
e )=e2, ne
is the electron collision frequency, w0 is the wave frequency, m and e are electron mass and charge respectively. The
parameter a stands for the degree of nonlocality of the nonlinear response. We prefer to keep a as a free parameter to
investigate the influence of the nonlocality on the properties of the multisolitons, though using an appropriate scaling
(see below) of y, q , x;y; and z one can get a = 1 in Eqs. (1) and (2). In the limit a2 À1, Eqs. (1) and (2)

Depending on the parameter l , we observed three different regimes of propagation, which are presented in the left
panel of Fig. 2 (for e = 0:02). The first regime corresponds to the region l <lcr, and we found lcr » 1:9. If l <lcr,
the initial dipole splits in two monopoles which move in the opposite directions without changing their shape and
without radiation, i. e. the monopoles just go away at infinity. This type of the evolution is shown in Fig. 1(a). The
splitting of the dipole into two monopoles can be easily understood. It is seen from Fig. 1 (left panel) that the bound
energy d N = Ndip¡2Nmon in the dipole tends to almost zero as l approaches lcr » 1:9. This explains why the dipole
with l ·lcr can be easily (i. e. under the action of extremely small initial perturbations) split into two monopole-type
solitons. A similar behavior, i.e. the decay of the initial dipole into two stable moving monopoles below some critical
value of l , was observed for the model with a Gaussian response function R(x ) in Eq. (5) [8].
The second regime corresponds to the region lcr <l <lth, where lth » 4. The numerical simulations clearly show
that in this range of the parameter l the dipoles are stable with respect to initial noisy perturbations. If the parameter
of perturbation e is not too large, the dipoles survive over huge distances (z > 3000). The stable propagation of the
dipole is illustrated in Figs. 2(b) (for l = 3:5 and e = 0:02).
The further (after lth »4) increasing the parameter l sharply shortens the propagation distances at which the dipole
survives, and, the dipoles with l >lth are unstable. The typical decay of the unstable dipole above the threshold value lth of the rescaled propagation constant is shown in Figs. 2(c). Thus, the stable dipoles exist only within a finite, rather
narrow range of the propagation constants l .
Figure 2 (right panel) illustrates the propagation of the tripole and quadrupole for l = 2, i.e. in the region, where
the dipole is stable. Generally speaking, the tripoles and quadrupoles turn out to be unstable, but for lcr < l < lth
they can survive on the quite considerable (compared to the characteristic diffraction length) distances. Tripoles and
quadrupoles with l <lcr decay into three and four monopoles respectively.
The dynamics of the dipole in our model Eqs. (1) and (2) is in sharp contrast to the dipole propagation in the model
with a Gaussian response function R(x ) in Eq. (5), where the stable dipoles were observed for all l >lth, where lth
is some threshold value [8]. A qualitatively different behavior of the dipoles in these models seems to be related to
the regularity properties of the functions R(x ) in Eq. (5) – for the model described by Eqs. (1) and (2) the function
R(x ) has a singularity at zero. The strong dependence of a stability criteria for multisoliton solutions on the regularity
properties of the kernel R in Eq. (5) was discussed also in [6]. One may present the following speculations. Suppose
that the exact solution of Eq. (5) is subjected to a small perturbation. Then, the shape of the self-consistent potential
well in Eq. (5) is distorted. One can see from Eq. (5), that for the slowly decaying perturbations, the variation dq of
the self-consistent potential can be anomalously large if the function R(x ) has a singularity. The fact that the stability
conditions essentially depend on the form of the nonlinear response function can also be illustrated in the following
manner. Consider the equation (5) with an arbitrary kernel R(x ). Following the standard procedure, one can obtain the
growth rate of the modulational instability (instability of the homogeneous field) g for the model Eq. (5)
g 2 = 2jY0j2k2 ˆR(k)¡k4; (11)
where jY0j2 is the intensity of the homogeneous field, ˆR(k) is the Fourier transform of the kernel R(x ). The model
described by Eqs. (1) and (2) corresponds to the function
k2+a2 : (12)
(k) = e¡ k2
4a2 ; (13)
we obtain the model with a Gaussian response function R(x ) [8]. In both cases, the parameter a stands for the degree
of nonlocality of the nonlinear response (a ! 0 corresponds to the strongly nonlocal regime). One can see that, for
the Gaussian model Eq. (13) the instability of perturbations with k2=(4a2) > 1 is exponentially suppressed, so that in
the strongly nonlocal regime (a ! 0, or large l in the rescaled variables) the instability is absent for (almost) all k
and jY0j2. The absolutely opposite situation takes place in the model with the response function Eq. (12) – strongly
nonlocal regime is the most favorable for the instability development and all modes with k4 < 2jY0j2=a2 are unstable.
These simple arguments can not, of course, rigorously explain the stability/instability of the multipole solutions. The
corresponding theoretical problem seems to be intriguing and highly nontrivial.
In conclusion, we have studied the bound states of the out-of-phase two-dimensional solitons in nonlocal nonlinear
media. We have demonstrated that stationary dipolar, tripolar and quadrupolar multisolitons may exist in the plasmas
with thermal nonlocal nonlinearity, if the beam power is above some critical value.We have investigated stability of the
multisolitons using direct numerical simulations. The tripoles and quadrupoles are found to be unstable with respect to
decay into fundamental solitons or fusion into single solitons depending on the beam power. At the same time we have
observed robust propagation of a dipolar multisoliton with moderate energy. Thus, our theoretical predictions open
the prospects for the experimental observations of the stable bound states of two out-of-phase wave beams in plasmas
with thermal self-focusing nonlocal nonlinearity.
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