104 research outputs found
Orbital stability of periodic waves in the class of reduced Ostrovsky equations
Periodic travelling waves are considered in the class of reduced Ostrovsky equations that describe low-frequency internal waves in the presence of rotation. The reduced Ostrovsky equations with either quadratic or cubic nonlinearities can be transformed to integrable equa- tions of the Klein–Gordon type by means of a change of coordinates. By using the conserved momentum and energy as well as an additional conserved quantity due to integrability, we prove that small-amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. The proof is based on construction of a Lyapunov functional, which is convex at the periodic wave and is conserved in the time evolution. We also show numerically that convexity of the Lyapunov functional holds for periodic waves of arbitrary amplitudes
Extinction of multiple shocks in the modular Burgers’ equation
We consider multiple shock waves in the Burgers’ equation with a modular advection term. It was previously shown that the modular Burgers’ equation admits a traveling viscous shock with a single interface, which is stable against smooth and exponentially localized perturbations. In contrast, we suggest in the present work with the help of energy estimates and numerical simulations that the evolution of shock waves with multiple interfaces leads to finite-time coalescence of two consecutive interfaces. We formulate a precise scaling law of the finite-time extinction supported by the interface equations and by numerical simulation
Moving gap solitons in periodic potentials
We address existence of moving gap solitons (traveling localized solutions)
in the Gross-Pitaevskii equation with a small periodic potential. Moving gap
solitons are approximated by the explicit localized solutions of the
coupled-mode system. We show however that exponentially decaying traveling
solutions of the Gross-Pitaevskii equation do not generally exist in the
presence of a periodic potential due to bounded oscillatory tails ahead and
behind the moving solitary waves. The oscillatory tails are not accounted in
the coupled-mode formalism and are estimated by using techniques of spatial
dynamics and local center-stable manifold reductions. Existence of bounded
traveling solutions of the Gross--Pitaevskii equation with a single bump
surrounded by oscillatory tails on a finite large interval of the spatial scale
is proven by using these technique. We also show generality of oscillatory
tails in other nonlinear equations with a periodic potential.Comment: 22 pages, 2 figure
Exact vortex solutions of the complex sine-Gordon theory on the plane
We construct explicit multivortex solutions for the first and second complex
sine-Gordon equations. The constructed solutions are expressible in terms of
the modified Bessel and rational functions, respectively. The vorticity-raising
and lowering Backlund transformations are interpreted as the Schlesinger
transformations of the fifth Painleve equation.Comment: 10 pages, 1 figur
Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential
We address a two-dimensional nonlinear elliptic problem with a
finite-amplitude periodic potential. For a class of separable symmetric
potentials, we study the bifurcation of the first band gap in the spectrum of
the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to
describe this bifurcation. The coupled-mode equations are derived by the
rigorous analysis based on the Fourier--Bloch decomposition and the Implicit
Function Theorem in the space of bounded continuous functions vanishing at
infinity. Persistence of reversible localized solutions, called gap solitons,
beyond the coupled-mode equations is proved under a non-degeneracy assumption
on the kernel of the linearization operator. Various branches of reversible
localized solutions are classified numerically in the framework of the
coupled-mode equations and convergence of the approximation error is verified.
Error estimates on the time-dependent solutions of the Gross--Pitaevskii
equation and the coupled-mode equations are obtained for a finite-time
interval.Comment: 32 pages, 16 figure
Stability of Spatial Optical Solitons
We present a brief overview of the basic concepts of the soliton stability
theory and discuss some characteristic examples of the instability-induced
soliton dynamics, in application to spatial optical solitons described by the
NLS-type nonlinear models and their generalizations. In particular, we
demonstrate that the soliton internal modes are responsible for the appearance
of the soliton instability, and outline an analytical approach based on a
multi-scale asymptotic technique that allows to analyze the soliton dynamics
near the marginal stability point. We also discuss some results of the rigorous
linear stability analysis of fundamental solitary waves and nonlinear impurity
modes. Finally, we demonstrate that multi-hump vector solitary waves may become
stable in some nonlinear models, and discuss the examples of stable
(1+1)-dimensional composite solitons and (2+1)-dimensional dipole-mode solitons
in a model of two incoherently interacting optical beams.Comment: 34 pages, 9 figures; to be published in: "Spatial Optical Solitons",
Eds. W. Torruellas and S. Trillo (Springer, New York
Gap solitons in Bose-Einstein condensates in linear and nonlinear optical lattices
Properties of localized states on array of BEC confined to a potential,
representing superposition of linear and nonlinear optical lattices are
investigated. For a shallow lattice case the coupled mode system has been
derived. The modulational instability of nonlinear plane waves is analyzed. We
revealed new types of gap solitons and studied their stability. For the first
time a moving soliton solution has been found. Analytical predictions are
confirmed by numerical simulations of the Gross-Pitaevskii equation with
jointly acting linear and nonlinear periodic potentials.Comment: 9 pages, 14 figure
An instability criterion for nonlinear standing waves on nonzero backgrounds
A nonlinear Schr\"odinger equation with repulsive (defocusing) nonlinearity
is considered. As an example, a system with a spatially varying coefficient of
the nonlinear term is studied. The nonlinearity is chosen to be repelling
except on a finite interval. Localized standing wave solutions on a non-zero
background, e.g., dark solitons trapped by the inhomogeneity, are identified
and studied. A novel instability criterion for such states is established
through a topological argument. This allows instability to be determined
quickly in many cases by considering simple geometric properties of the
standing waves as viewed in the composite phase plane. Numerical calculations
accompany the analytical results.Comment: 20 pages, 11 figure
Bifurcations and stability of gap solitons in periodic potentials
We analyze the existence, stability, and internal modes of gap solitons in
nonlinear periodic systems described by the nonlinear Schrodinger equation with
a sinusoidal potential, such as photonic crystals, waveguide arrays,
optically-induced photonic lattices, and Bose-Einstein condensates loaded onto
an optical lattice. We study bifurcations of gap solitons from the band edges
of the Floquet-Bloch spectrum, and show that gap solitons can appear near all
lower or upper band edges of the spectrum, for focusing or defocusing
nonlinearity, respectively. We show that, in general, two types of gap solitons
can bifurcate from each band edge, and one of those two is always unstable. A
gap soliton corresponding to a given band edge is shown to possess a number of
internal modes that bifurcate from all band edges of the same polarity. We
demonstrate that stability of gap solitons is determined by location of the
internal modes with respect to the spectral bands of the inverted spectrum and,
when they overlap, complex eigenvalues give rise to oscillatory instabilities
of gap solitons.Comment: 18 pages, 11 figures; updated bibliograph
Breakup of self-guided light beams into X wave trains
Relaxation of the nonlinear spatiotemporal dynamics of cylindrically symmetric SchrÄodinger solitons due to their temporal modulation instability leads to soliton break-up into a train of X waves
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