218 research outputs found
Two-soliton collisions in a near-integrable lattice system
We examine collisions between identical solitons in a weakly perturbed
Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which
corresponds to the Salerno model, and may be realized as an array of strongly
overlapping nonlinear optical waveguides), or a quintic perturbation, or both.
Complex dependences of the outcomes of the collisions on the initial phase
difference between the solitons and location of the collision point are
observed. Large changes of amplitudes and velocities of the colliding solitons
are generated by weak perturbations, showing that the elasticity of soliton
collisions in the AL model is fragile (for instance, the Salerno's perturbation
with the relative strength of 0.08 can give rise to a change of the solitons'
amplitudes by a factor exceeding 2). Exact and approximate conservation laws in
the perturbed system are examined, with a conclusion that the small
perturbations very weakly affect the norm and energy conservation, but
completely destroy the conservation of the lattice momentum, which is explained
by the absence of the translational symmetry in generic nonintegrable lattice
models. Data collected for a very large number of collisions correlate with
this conclusion. Asymmetry of the collisions (which is explained by the
dependence on the location of the central point of the collision relative to
the lattice, and on the phase difference between the solitons) is investigated
too, showing that the nonintegrability-induced effects grow almost linearly
with the perturbation strength. Different perturbations (cubic and quintic
ones) produce virtually identical collision-induced effects, which makes it
possible to compensate them, thus finding a special perturbed system with
almost elastic soliton collisions.Comment: Phys. Rev. E, in pres
Interactions and Collisions of Discrete Breathers in Two-Species Bose-Einstein Condensates in Optical Lattices
The dynamics of static and travelling breathers in two-species Bose-Einstein
condensates in a one-dimensional optical lattice is modelled within the
tight-binding approximation. Two coupled discrete nonlinear Schr\"odinger
equations describe the interaction of the condensates in two cases of
relevance: a mixture of two ytterbium isotopes and a mixture of Rb and
K. Depending on their initial separation, interaction between static
breathers of different species can lead to the formation of symbiotic
structures and transform one of the breathers from a static into a travelling
one. Collisions between travelling and static discrete breathers composed of
different species are separated in four distinct regimes ranging from totally
elastic when the interspecies interaction is highly attractive to mutual
destruction when the interaction is sufficiently large and repulsive. We
provide an explanation of the collision features in terms of the interspecies
coupling and the negative effective mass of the discrete breathers.Comment: 11 pages, 10 figure
Moving and colliding pulses in the subcritical Ginzburg-Landau model with a standing-wave drive
We show the existence of steadily moving solitary pulses (SPs) in the complex
Ginzburg-Landau (CGL) equation, which includes the cubic-quintic (CQ)
nonlinearity and a conservative linear driving term, whose amplitude is a
standing wave with wavenumber and frequency , the motion of the
SPs being possible at velocities , which provide locking to the
drive. A realization of the model may be provided by traveling-wave convection
in a narrow channel with a standing wave excited in its bottom (or on the
surface). An analytical approximation is developed, based on an effective
equation of motion for the SP coordinate. Direct simulations demonstrate that
the effective equation accurately predicts characteristics of the driven motion
of pulses, such as a threshold value of the drive's amplitude. Collisions
between two solitons traveling in opposite directions are studied by means of
direct simulations, which reveal that they restore their original shapes and
velocity after the collision.Comment: 7 pages, 5 eps figure
Exact Localized Solutions of Quintic Discrete Nonlinear Schr\"odinger Equation
We study a new quintic discrete nonlinear Schr\"odinger (QDNLS) equation
which reduces naturally to an interesting symmetric difference equation of the
form . Integrability of the symmetric mapping
is checked by singularity confinement criteria and growth properties. Some new
exact localized solutions for integrable cases are presented for certain sets
of parameters. Although these exact localized solutions represent only a small
subset of the large variety of possible solutions admitted by the QDNLS
equation, those solutions presented here are the first example of exact
localized solutions of the QDNLS equation. We also find chaotic behavior for
certain parameters of nonintegrable case.Comment: 12 pages,4 figures(eps files),revised,Physics Letters A, In pres
Motion of discrete solitons assisted by nonlinearity management
We demonstrate that periodic modulation of the nonlinearity coefficient in
the discrete nonlinear Schr\"{o}dinger (DNLS) equation can strongly facilitate
creation of traveling solitons in the lattice. We predict this possibility in
an analytical form, and test it in direct simulations. Systematic simulations
reveal several generic dynamical regimes, depending on the amplitude and
frequency of the time modulation, and on initial thrust which sets the soliton
in motion. These regimes include irregular motion, regular motion of a decaying
soliton, and regular motion of a stable one. The motion may occur in both the
straight and reverse directions, relative to the initial thrust. In the case of
stable motion, extremely long simulations in a lattice with periodic boundary
conditions demonstrate that the soliton keeps moving as long as we can monitor
without any visible loss. Velocities of moving stable solitons are in good
agreement with the analytical prediction, which is based on requiring a
resonance between the ac drive and motion of the soliton through the periodic
potential. All the generic dynamical regimes are mapped in the model's
parameter space. Collisions between moving stable solitons are briefly
investigated too, with a conclusion that two different outcomes are possible:
elastic bounce, or bounce with mass transfer from one soliton to the other. The
model can be realized experimentally in a Bose-Einstein condensate trapped in a
deep optical lattice
Multistable Solitons in Higher-Dimensional Cubic-Quintic Nonlinear Schroedinger Lattices
We study the existence, stability, and mobility of fundamental discrete
solitons in two- and three-dimensional nonlinear Schroedinger lattices with a
combination of cubic self-focusing and quintic self-defocusing onsite
nonlinearities. Several species of stationary solutions are constructed, and
bifurcations linking their families are investigated using parameter
continuation starting from the anti-continuum limit, and also with the help of
a variational approximation. In particular, a species of hybrid solitons,
intermediate between the site- and bond-centered types of the localized states
(with no counterpart in the 1D model), is analyzed in 2D and 3D lattices. We
also discuss the mobility of multi-dimensional discrete solitons that can be
set in motion by lending them kinetic energy exceeding the appropriately
crafted Peierls-Nabarro barrier; however, they eventually come to a halt, due
to radiation loss.Comment: 12 pages, 17 figure
Discrete Nonlinear Schrodinger Equations with arbitrarily high order nonlinearities
A class of discrete nonlinear Schrodinger equations with arbitrarily high
order nonlinearities is introduced. These equations are derived from the same
Hamiltonian using different Poisson brackets and include as particular cases
the saturable discrete nonlinear Schrodinger equation and the Ablowitz-Ladik
equation. As a common property, these equations possess three kinds of exact
analytical stationary solutions for which the Peierls-Nabarro barrier is zero.
Several properties of these solutions, including stability, discrete breathers
and moving solutions, are investigated
Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity
We investigate mobility regimes for localized modes in the discrete nonlinear
Schr\"{o}dinger (DNLS) equation with the cubic-quintic onsite terms. Using the
variational approximation (VA), the largest soliton's total power admitting
progressive motion of kicked discrete solitons is predicted, by comparing the
effective kinetic energy with the respective Peierls-Nabarro (PN) potential
barrier. The prediction is novel for the DNLS model with the cubic-only
nonlinearity too, demonstrating a reasonable agreement with numerical findings.
Small self-focusing quintic term quickly suppresses the mobility. In the case
of the competition between the cubic self-focusing and quintic self-defocusing
terms, we identify parameter regions where odd and even fundamental modes
exchange their stability, involving intermediate asymmetric modes. In this
case, stable solitons can be set in motion by kicking, so as to let them pass
the PN barrier. Unstable solitons spontaneously start oscillatory or
progressive motion, if they are located, respectively, below or above a
mobility threshold. Collisions between moving discrete solitons, at the
competing nonlinearities frame, are studied too.Comment: 12 pages, 15 figure
- …