41 research outputs found
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
Coupled-mode systems are used in physical literature to simplify the
nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic
potential and to approximate localized solutions called gap solitons by
analytical expressions involving hyperbolic functions. We justify the use of
the one-dimensional stationary coupled-mode system for a relevant elliptic
problem by employing the method of Lyapunov--Schmidt reductions in Fourier
space. In particular, existence of periodic/anti-periodic and decaying
solutions is proved and the error terms are controlled in suitable norms. The
use of multi-dimensional stationary coupled-mode systems is justified for
analysis of bifurcations of periodic/anti-periodic solutions in a small
multi-dimensional periodic potential.Comment: 18 pages, no figure
Stable vortex and dipole vector solitons in a saturable nonlinear medium
We study both analytically and numerically the existence, uniqueness, and
stability of vortex and dipole vector solitons in a saturable nonlinear medium
in (2+1) dimensions. We construct perturbation series expansions for the vortex
and dipole vector solitons near the bifurcation point where the vortex and
dipole components are small. We show that both solutions uniquely bifurcate
from the same bifurcation point. We also prove that both vortex and dipole
vector solitons are linearly stable in the neighborhood of the bifurcation
point. Far from the bifurcation point, the family of vortex solitons becomes
linearly unstable via oscillatory instabilities, while the family of dipole
solitons remains stable in the entire domain of existence. In addition, we show
that an unstable vortex soliton breaks up either into a rotating dipole soliton
or into two rotating fundamental solitons.Comment: To appear in Phys. Rev.
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
Instabilities of dispersion-managed solitons in the normal dispersion regime
Dispersion-managed solitons are reviewed within a Gaussian variational
approximation and an integral evolution model. In the normal regime of the
dispersion map (when the averaged path dispersion is negative), there are two
solitons of different pulse duration and energy at a fixed propagation
constant. We show that the short soliton with a larger energy is linearly
(exponentially) unstable. The other (long) soliton with a smaller energy is
linearly stable but hits a resonance with excitations of the dispersion map.
The results are compared with the results from the recent publicationsComment: 20 figures, 20 pages. submitted to Phys. Rev.
Bifurcations of self-similar solutions for reversing interfaces in the slow diffusion equation with strong absorption
Bifurcations of self-similar solutions for reversing interfaces are studied in the slow diffusion equation with strong absorption. The self-similar solutions bifurcate from the time-independent solutions for standing interfaces. We show that such bifurcations occur at particular points in parameter space (characterizing the exponents in the diffusion and absorption terms) where the confluent hypergeometric functions satisfying Kummer's differential equation truncate to finite polynomials. A two-scale asymptotic method is employed to obtain the local dependencies of the self-similar reversing interfaces near the bifurcation points. The asymptotic results are shown to be in excellent agreement with numerical approximations of the self-similar solutions
Periodic Travelling Waves in Dimer Granular Chains
We study bifurcations of periodic travelling waves in granular dimer chains
from the anti-continuum limit, when the mass ratio between the light and heavy
beads is zero. We show that every limiting periodic wave is uniquely continued
with respect to the mass ratio parameter and the periodic waves with the
wavelength larger than a certain critical value are spectrally stable.
Numerical computations are developed to study how this solution family is
continued to the limit of equal mass ratio between the beads, where periodic
travelling waves of granular monomer chains exist
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
Two-color nonlinear localized photonic modes
We analyze second-harmonic generation (SHG) at a thin effectively quadratic
nonlinear interface between two linear optical media. We predict multistability
of SHG for both plane and localized waves, and also describe two-color
localized photonic modes composed of a fundamental wave and its second harmonic
coupled together by parametric interaction at the interface.Comment: 4 pages, 5 figures (updated references
Instabilities in the two-dimensional cubic nonlinear Schrodinger equation
The two-dimensional cubic nonlinear Schrodinger equation (NLS) can be used as
a model of phenomena in physical systems ranging from waves on deep water to
pulses in optical fibers. In this paper, we establish that every
one-dimensional traveling wave solution of NLS with trivial phase is unstable
with respect to some infinitesimal perturbation with two-dimensional structure.
If the coefficients of the linear dispersion terms have the same sign then the
only unstable perturbations have transverse wavelength longer than a
well-defined cut-off. If the coefficients of the linear dispersion terms have
opposite signs, then there is no such cut-off and as the wavelength decreases,
the maximum growth rate approaches a well-defined limit.Comment: 4 pages, 4 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