139 research outputs found
Discrete solitons in PT-symmetric lattices
We prove existence of discrete solitons in infinite parity-time (PT-)
symmetric lattices by means of analytical continuation from the anticontinuum
limit. The energy balance between dissipation and gain implies that in the
anticontinuum limit the solitons are constructed from elementary PT-symmetric
blocks such as dimers, quadrimers, or more general oligomers. We consider in
detail a chain of coupled dimers, analyze bifurcations of discrete solitons
from the anticontinuum limit and show that the solitons are stable in a
sufficiently large region of the lattice parameters. The generalization of the
approach is illustrated on two examples of networks of quadrimers, for which
stable discrete solitons are also found.Comment: 6 pages, 6 figures; accepted to EPL, www.epletters.ne
Stability of localized modes in PT-symmetric nonlinear potentials
We report on detailed investigation of the stability of localized modes in
the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT)
symmetric potential. We are particularly focusing on the case where the
spatially-dependent nonlinearity is purely imaginary. We compute the Evans
function of the linear operator determining the linear stability of localized
modes. Results of the Evans function analysis predict that for sufficiently
small dissipation localized modes become stable when the propagation constant
exceeds certain threshold value. This is the case for periodic and
-shaped complex potentials where the modes having widths comparable with
or smaller than the characteristic width of the complex potential are stable,
while broad modes are unstable. In contrast, in complex potentials that change
linearly with transverse coordinate all modes are stable, what suggests that
the relation between width of the modes and spatial size of the complex
potential define the stability in the general case. These results were
confirmed using the direct propagation of the solutions for the mentioned
examples.Comment: 6 pages, 4 figures; accepted to Europhysics Letters,
https://www.epletters.net
Small-amplitude excitations in a deformable discrete nonlinear Schroedinger equation
A detailed analysis of the small-amplitude solutions of a deformed discrete
nonlinear Schr\"{o}dinger equation is performed. For generic deformations the
system possesses "singular" points which split the infinite chain in a number
of independent segments. We show that small-amplitude dark solitons in the
vicinity of the singular points are described by the Toda-lattice equation
while away from the singular points are described by the Korteweg-de Vries
equation. Depending on the value of the deformation parameter and of the
background level several kinds of solutions are possible. In particular we
delimit the regions in the parameter space in which dark solitons are stable in
contrast with regions in which bright pulses on nonzero background are
possible. On the boundaries of these regions we find that shock waves and
rapidly spreading solutions may exist.Comment: 18 pages (RevTex), 13 figures available upon reques
Nonlinear Modulation of Multi-Dimensional Lattice Waves
The equations governing weakly nonlinear modulations of -dimensional
lattices are considered using a quasi-discrete multiple-scale approach. It is
found that the evolution of a short wave packet for a lattice system with cubic
and quartic interatomic potentials is governed by generalized Davey-Stewartson
(GDS) equations, which include mean motion induced by the oscillatory wave
packet through cubic interatomic interaction. The GDS equations derived here
are more general than those known in the theory of water waves because of the
anisotropy inherent in lattices. Generalized Kadomtsev-Petviashvili equations
describing the evolution of long wavelength acoustic modes in two and three
dimensional lattices are also presented. Then the modulational instability of a
-dimensional Stokes lattice wave is discussed based on the -dimensional
GDS equations obtained. Finally, the one- and two-soliton solutions of
two-dimensional GDS equations are provided by means of Hirota's bilinear
transformation method.Comment: Submitted to PR
Effected of Feshbach resonance on dynamics of matter waves in optical lattices
Mean-filed dynamics of a Bose-Einstein condensate (BEC) loaded in an optical
lattice (OL), confined by a parabolic potentials, and subjected to change of a
scattering length by means of the Feshbach resonance (FR), is considered. The
system is described by the Gross-Pitaevskii (GP) equation with varying
nonlinearity, which in a number of cases can be reduced a one-dimensional
perturbed nonlinear Schr\"{o}dinger (NLS) equation. A particular form of the
last one depends on relations among BEC parameters. We describe periodic
solutions of the NLS equation and their adiabatic dynamics due to varying
nonlinearity; carry out numerical study of the dynamics of the NLS equation
with periodic and parabolic trap potentials. We pay special attention to
processes of generation of trains of bright and dark matter solitons from
initially periodic waves.Comment: 16 pages, 11 figures (revised version). to be published in Phys. Rev.
A (2005
On dissipationless shock waves in a discrete nonlinear Schr\"odinger equation
It is shown that the generalized discrete nonlinear Schr\"odinger equation
can be reduced in a small amplitude approximation to the KdV, mKdV, KdV(2) or
the fifth-order KdV equations, depending on values of the parameters. In
dispersionless limit these equations lead to wave breaking phenomenon for
general enough initial conditions, and, after taking into account small
dispersion effects, result in formation of dissipationless shock waves. The
Whitham theory of modulations of nonlinear waves is used for analytical
description of such waves.Comment: 15 pages, 9 figure
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