13 research outputs found
Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation
We consider the nonlinear Schr\"{o}dinger equation , with and and with periodic in each coordinate direction. This problem
describes the interface of two periodic media, e.g. photonic crystals. We study
the existence of ground state solutions (surface gap soliton ground
states) for . Using a concentration compactness
argument, we provide an abstract criterion for the existence based on ground
state energies of each periodic problem (with and ) as well as a more practical
criterion based on ground states themselves. Examples of interfaces satisfying
these criteria are provided. In 1D it is shown that, surprisingly, the criteria
can be reduced to conditions on the linear Bloch waves of the operators
and .Comment: definition of ground and bound states added, assumption (H2) weakened
(sign changing nonlinearity is now allowed); 33 pages, 4 figure
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
Doubly periodic waves of a discrete nonlinear Schr\"odinger system with saturable nonlinearity
A system of two discrete nonlinear Schr\"odinger equations of the
Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly
periodic wave, whose long wave limit is also derived. As a by-product, several
new solutions of the elliptic type are provided for NLS-type discrete and
continuous systems.Comment: 12 pages, to appear, Journal of nonlinear mathematical physic
Stability of dark solitons in a Bose-Einstein condensate trapped in an optical lattice
We investigate the stability of dark solitons (DSs) in an effectively
one-dimensional Bose-Einstein condensate in the presence of the magnetic
parabolic trap and an optical lattice (OL). The analysis is based on both the
full Gross-Pitaevskii equation and its tight-binding approximation counterpart
(discrete nonlinear Schr{\"o}dinger equation). We find that DSs are subject to
weak instabilities with an onset of instability mainly governed by the period
and amplitude of the OL. The instability, if present, sets in at large times
and it is characterized by quasi-periodic oscillations of the DS about the
minimum of the parabolic trap.Comment: Typo fixed in Eq. (1): cos^2 -> sin^
Bessel X waves in two- and three-dimensional bidispersive optical systems
We show that new families of two- and three-dimensional nondiffracting Bessel X waves are possible in linear bidispersive optical systems. These X waves can be observed in both bulk and waveguide configurations as well as in photonic crystal lattices that simultaneously exhibit normal and anomalous dispersive-diffractive properties in different spatial or spatiotemporal coordinates. © 2004 Optical Society of America
Three-dimensional vortex solitons in self-defocusing media
The existence and robustness of dark vortices in bi-dispersive and/or normally dispersive self-defocusing nonlinear media is demonstrated. The underlying equation is the bi-dispersive three-dimensional nonlinear Schrodinger equation. These solutions can be considered as extensions of two-dimensional dark vortex solitons which, along the third dimension, remain localized due to the interplay between diffraction and nonlinearity. Such vortex solitons can be observed in optical media with normal dispersion, normal diffraction, and defocusing nonlinearity