5,091 research outputs found
Statistical Description of Acoustic Turbulence
We develop expressions for the nonlinear wave damping and frequency
correction of a field of random, spatially homogeneous, acoustic waves. The
implications for the nature of the equilibrium spectral energy distribution are
discussedComment: PRE, Submitted. REVTeX, 16 pages, 3 figures (not included) PS Source
of the paper with figures avalable at
http://lvov.weizmann.ac.il/onlinelist.htm
Nonlinear Lattice Dynamics of Bose-Einstein Condensates
The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine
thermalization in non-metallic solids and develop ``experimental'' techniques
for studying nonlinear problems, continues to yield a wealth of results in the
theory and applications of nonlinear Hamiltonian systems with many degrees of
freedom. Inspired by the studies of this seminal model, solitary-wave dynamics
in lattice dynamical systems have proven vitally important in a diverse range
of physical problems--including energy relaxation in solids, denaturation of
the DNA double strand, self-trapping of light in arrays of optical waveguides,
and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular,
due to their widely ranging and easily manipulated dynamical apparatuses--with
one to three spatial dimensions, positive-to-negative tuning of the
nonlinearity, one to multiple components, and numerous experimentally
accessible external trapping potentials--provide one of the most fertile
grounds for the analysis of solitary waves and their interactions. In this
paper, we review recent research on BECs in the presence of deep periodic
potentials, which can be reduced to nonlinear chains in appropriate
circumstances. These reductions, in turn, exhibit many of the remarkable
nonlinear structures (including solitons, intrinsic localized modes, and
vortices) that lie at the heart of the nonlinear science research seeded by the
FPU paradigm.Comment: 10 pages, revtex, two-columns, 3 figs, accepted fpr publication in
Chaos's focus issue on the 50th anniversary of the publication of the
Fermi-Pasta-Ulam problem; minor clarifications (and a couple corrected typos)
from previous versio
Kinetic equation for a dense soliton gas
We propose a general method to derive kinetic equations for dense soliton
gases in physical systems described by integrable nonlinear wave equations. The
kinetic equation describes evolution of the spectral distribution function of
solitons due to soliton-soliton collisions. Owing to complete integrability of
the soliton equations, only pairwise soliton interactions contribute to the
solution and the evolution reduces to a transport of the eigenvalues of the
associated spectral problem with the corresponding soliton velocities modified
by the collisions. The proposed general procedure of the derivation of the
kinetic equation is illustrated by the examples of the Korteweg -- de Vries
(KdV) and nonlinear Schr\"odinger (NLS) equations. As a simple physical example
we construct an explicit solution for the case of interaction of two cold NLS
soliton gases.Comment: 4 pages, 1 figure, final version published in Phys. Rev. Let
Hopf algebras and characters of classical groups
Schur functions provide an integral basis of the ring of symmetric functions.
It is shown that this ring has a natural Hopf algebra structure by identifying
the appropriate product, coproduct, unit, counit and antipode, and their
properties. Characters of covariant tensor irreducible representations of the
classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur
functions, and the Hopf algebra is exploited in the determination of
group-subgroup branching rules and the decomposition of tensor products. The
analysis is carried out in terms of n-independent universal characters. The
corresponding rings, CharGL, CharO and CharSp, of universal characters each
have their own natural Hopf algebra structure. The appropriate product,
coproduct, unit, counit and antipode are identified in each case.Comment: 9 pages. Uses jpconf.cls and jpconf11.clo. Presented by RCK at
SSPCM'07, Myczkowce, Poland, Sept 200
Solitary vortex couples in viscoelastic Couette flow
We report experimental observation of a localized structure, which is of a
new type for dissipative systems. It appears as a solitary vortex couple
("diwhirl") in Couette flow with highly elastic polymer solutions. A unique
property of the diwhirls is that they are stationary, in contrast to the usual
localized wave structures in both Hamiltonian and dissipative systems which are
stabilized by wave dispersion. It is also a new object in fluid dynamics - a
couple of vortices that build a single entity somewhat similar to a magnetic
dipole. The diwhirls arise as a result of a purely elastic instability through
a hysteretic transition at negligible Reynolds numbers. It is suggested that
the vortex flow is driven by the same forces that cause the Weissenberg effect.
The diwhirls have a striking asymmetry between the inflow and outflow, which is
also an essential feature of the suggested elastic instability mechanism.Comment: 9 pages (LaTeX), 5 Postscript figures, submitte
"Doubled" generalized Landau-Lifshiz hierarchies and special quasigraded Lie algebras
Using special quasigraded Lie algebras we obtain new hierarchies of
integrable nonlinear vector equations admitting zero-curvature representations.
Among them the most interesting is extension of the generalized Landau-Lifshitz
hierarchy which we call "doubled" generalized Landau-Lifshiz hierarchy. This
hierarchy can be also interpreted as an anisotropic vector generalization of
"modified" Sine-Gordon hierarchy or as a very special vector generalization of
so(3) anisotropic chiral field hierarchy.Comment: 16 pages, no figures, submitted to Journal of Physics
Multi-Phase Patterns in Periodically Forced Oscillatory Systems
Periodic forcing of an oscillatory system produces frequency locking bands
within which the system frequency is rationally related to the forcing
frequency. We study extended oscillatory systems that respond to uniform
periodic forcing at one quarter of the forcing frequency (the 4:1 resonance).
These systems possess four coexisting stable states, corresponding to uniform
oscillations with successive phase shifts of . Using an amplitude
equation approach near a Hopf bifurcation to uniform oscillations, we study
front solutions connecting different phase states. These solutions divide into
two groups: -fronts separating states with a phase shift of and
-fronts separating states with a phase shift of . We find a new
type of front instability where a stationary -front ``decomposes'' into a
pair of traveling -fronts as the forcing strength is decreased. The
instability is degenerate for an amplitude equation with cubic nonlinearities.
At the instability point a continuous family of pair solutions exists,
consisting of -fronts separated by distances ranging from zero to
infinity. Quintic nonlinearities lift the degeneracy at the instability point
but do not change the basic nature of the instability. We conjecture the
existence of similar instabilities in higher 2n:1 resonances (n=3,4,..) where
stationary -fronts decompose into n traveling -fronts. The
instabilities designate transitions from stationary two-phase patterns to
traveling 2n-phase patterns. As an example, we demonstrate with a numerical
solution the collapse of a four-phase spiral wave into a stationary two-phase
pattern as the forcing strength within the 4:1 resonance is increased
New way to achieve chaotic synchronization in spatially extended systems
We study the spatio-temporal behavior of simple coupled map lattices with
periodic boundary conditions. The local dynamics is governed by two maps,
namely, the sine circle map and the logistic map respectively. It is found that
even though the spatial behavior is irregular for the regularly coupled
(nearest neighbor coupling) system, the spatially synchronized (chaotic
synchronization) as well as periodic solution may be obtained by the
introduction of three long range couplings at the cost of three nearest
neighbor couplings.Comment: 5 pages (revtex), 7 figures (eps, included
KP solitons in shallow water
The main purpose of the paper is to provide a survey of our recent studies on
soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The
classification is based on the far-field patterns of the solutions which
consist of a finite number of line-solitons. Each soliton solution is then
defined by a point of the totally non-negative Grassmann variety which can be
parametrized by a unique derangement of the symmetric group of permutations.
Our study also includes certain numerical stability problems of those soliton
solutions. Numerical simulations of the initial value problems indicate that
certain class of initial waves asymptotically approach to these exact solutions
of the KP equation. We then discuss an application of our theory to the Mach
reflection problem in shallow water. This problem describes the resonant
interaction of solitary waves appearing in the reflection of an obliquely
incident wave onto a vertical wall, and it predicts an extra-ordinary four-fold
amplification of the wave at the wall. There are several numerical studies
confirming the prediction, but all indicate disagreements with the KP theory.
Contrary to those previous numerical studies, we find that the KP theory
actually provides an excellent model to describe the Mach reflection phenomena
when the higher order corrections are included to the quasi-two dimensional
approximation. We also present laboratory experiments of the Mach reflection
recently carried out by Yeh and his colleagues, and show how precisely the KP
theory predicts this wave behavior.Comment: 50 pages, 25 figure
Kink Arrays and Solitary Structures in Optically Biased Phase Transition
An interphase boundary may be immobilized due to nonlinear diffractional
interactions in a feedback optical device. This effect reminds of the Turing
mechanism, with the optical field playing the role of a diffusive inhibitor.
Two examples of pattern formation are considered in detail: arrays of kinks in
1d, and solitary spots in 2d. In both cases, a large number of equilibrium
solutions is possible due to the oscillatory character of diffractional
interaction.Comment: RevTeX 13 pages, 3 PS-figure
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