109 research outputs found
Effects of finite curvature on soliton dynamics in a chain of nonlinear oscillators
We consider a curved chain of nonlinear oscillators and show that the
interplay of curvature and nonlinearity leads to a number of qualitative
effects. In particular, the energy of nonlinear localized excitations centered
on the bending decreases when curvature increases, i.e. bending manifests
itself as a trap for excitations. Moreover, the potential of this trap is
double-well, thus leading to a symmetry breaking phenomenon: a symmetric
stationary state may become unstable and transform into an energetically
favorable asymmetric stationary state. The essentials of symmetry breaking are
examined analytically for a simplified model. We also demonstrate a threshold
character of the scattering process, i.e. transmission, trapping, or reflection
of the moving nonlinear excitation passing through the bending.Comment: 13 pages (LaTeX) with 10 figures (EPS
Soliton dynamics in damped and forced Boussinesq equations
We investigate the dynamics of a lattice soliton on a monatomic chain in the
presence of damping and external forces. We consider Stokes and hydrodynamical
damping. In the quasi-continuum limit the discrete system leads to a damped and
forced Boussinesq equation. By using a multiple-scale perturbation expansion up
to second order in the framework of the quasi-continuum approach we derive a
general expression for the first-order velocity correction which improves
previous results. We compare the soliton position and shape predicted by the
theory with simulations carried out on the level of the monatomic chain system
as well as on the level of the quasi-continuum limit system. For this purpose
we restrict ourselves to specific examples, namely potentials with cubic and
quartic anharmonicities as well as the truncated Morse potential, without
taking into account external forces. For both types of damping we find a good
agreement with the numerical simulations both for the soliton position and for
the tail which appears at the rear of the soliton. Moreover we clarify why the
quasi-continuum approximation is better in the hydrodynamical damping case than
in the Stokes damping case
Curvature-induced symmetry breaking in nonlinear Schrodinger models
We consider a curved chain of nonlinear oscillators and show that the
interplay of curvature and nonlinearity leads to a symmetry breaking when an
asymmetric stationary state becomes energetically more favorable than a
symmetric stationary state. We show that the energy of localized states
decreases with increasing curvature, i.e. bending is a trap for nonlinear
excitations. A violation of the Vakhitov-Kolokolov stability criterium is found
in the case where the instability is due to the softening of the Peierls
internal mode.Comment: 4 pages (LaTex) with 6 figures (EPS
Solitons in anharmonic chains with ultra-long-range interatomic interactions
We study the influence of long-range interatomic interactions on the
properties of supersonic pulse solitons in anharmonic chains. We show that in
the case of ultra-long-range (e.g., screened Coulomb) interactions three
different types of pulse solitons coexist in a certain velocity interval: one
type is unstable but the two others are stable. The high-energy stable soliton
is broad and can be described in the quasicontinuum approximation. But the
low-energy stable soliton consists of two components, short-range and
long-range ones, and can be considered as a bound state of these components.Comment: 4 pages (LaTeX), 5 figures (Postscript); submitted to Phys. Rev.
Effects of spin-elastic interactions in frustrated Heisenberg antiferromagnets
The Heisenberg antiferromagnet on a compressible triangular lattice in the
spin- wave approximation is considered. It is shown that the interaction
between quantum fluctuations and elastic degrees of freedom stabilizes the low
symmetric L-phase with a collinear Neel magnetic ordering. Multi-stability in
the dependence of the on-site magnetization on an unaxial pressure is found.Comment: Revtex, 4 pages, 2 eps figure
Localization of nonlinear excitations in curved waveguides
Motivated by the example of a curved waveguide embedded in a photonic
crystal, we examine the effects of geometry in a ``quantum channel'' of
parabolic form. We study the linear case and derive exact as well as
approximate expressions for the eigenvalues and eigenfunctions of the linear
problem. We then proceed to the nonlinear setting and its stationary states in
a number of limiting cases that allow for analytical treatment. The results of
our analysis are used as initial conditions in direct numerical simulations of
the nonlinear problem and localized excitations are found to persist, as well
as to have interesting relaxational dynamics. Analogies of the present problem
in contexts related to atomic physics and particularly to Bose-Einstein
condensation are discussed.Comment: 14 pages, 4 figure
Kink propagation in a two-dimensional curved Josephson junction
We consider the propagation of sine-Gordon kinks in a planar curved strip as
a model of nonlinear wave propagation in curved wave guides. The homogeneous
Neumann transverse boundary conditions, in the curvilinear coordinates, allow
to assume a homogeneous kink solution. Using a simple collective variable
approach based on the kink coordinate, we show that curved regions act as
potential barriers for the wave and determine the threshold velocity for the
kink to cross. The analysis is confirmed by numerical solution of the 2D
sine-Gordon equation.Comment: 8 pages, 4 figures (2 in color
On the continuum limit for discrete NLS with long-range lattice interactions
We consider a general class of discrete nonlinear Schroedinger equations
(DNLS) on the lattice with mesh size . In the continuum
limit when , we prove that the limiting dynamics are given by a
nonlinear Schroedinger equation (NLS) on with the fractional
Laplacian as dispersive symbol. In particular, we obtain
that fractional powers arise from long-range lattice
interactions when passing to the continuum limit, whereas NLS with the
non-fractional Laplacian describes the dispersion in the continuum
limit for short-range lattice interactions (e.g., nearest-neighbor
interactions).
Our results rigorously justify certain NLS model equations with fractional
Laplacians proposed in the physics literature. Moreover, the arguments given in
our paper can be also applied to discuss the continuum limit for other lattice
systems with long-range interactions.Comment: 26 pages; no figures. Some minor revisions. To appear in Comm. Math.
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