Modulational Instability in Bose-Einstein Condensates under Feshbach Resonance Management

We investigate the modulational instability of nonlinear Schr{\"o}dinger equations with periodic variation of their coefficients. In particular, we focus on the case of the recently proposed, experimentally realizable protocol of Feshbach Resonance Management for Bose-Einstein condensates. We derive the corresponding linear stability equation analytically and we show that it can be reduced to a Kronig-Penney model, which allows the determination of the windows of instability. The results are tested numerically in the absence, as well as in the presence of the magnetic trapping potential

Spinor Bose-Einstein condensates in double-well potentials

We consider the statics and dynamics of F = 1 spinor Bose–Einstein condensates (BECs) confined in double-well potentials. We use a two-mode Galerkin-type quasi-analytical approximation to describe the stationary states of the system. This way, we are able to obtain not only earlier results based on the single-mode approximation (SMA) frequently used in studies of spinor BECs, but also additional modes that involve either two or all three spinor components of the F = 1 spinor BEC. The results based on this Galerkin-type decomposition are in good agreement with the analysis of the full system. We subsequently analyze the stability of these multi-component states, as well as their dynamics when we find them to be unstable. The instabilities of the symmetric or anti-symmetric states exhibit symmetry-breaking and recurrent asymmetric patterns. Our results yield qualitatively similar bifurcation diagrams both for polar (such as 23Na) and ferromagnetic (such as 87Rb) spinor BECs

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

Theory of Multidimensional Solitons

We review a number of topics germane to higher-dimensional solitons in Bose-Einstein condensates. For dark solitons, we discuss dark band and planar solitons; ring dark solitons and spherical shell solitons; solitary waves in restricted geometries; vortex rings and rarefaction pulses; and multi-component Bose-Einstein condensates. For bright solitons, we discuss instability, stability, and metastability; bright soliton engineering, including pulsed atom lasers; solitons in a thermal bath; soliton-soliton interactions; and bright ring solitons and quantum vortices. A thorough reference list is included.Comment: review paper, to appear as Chapter 5a in "Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment," edited by P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonzalez (Springer-Verlag

Weak topological optical solitons in the femtosecond time scale

Perturbative topological solitary-wave solutions of a generalized nonlinear Schrodinger equation, describing optical propagation in the femtosecond time scale, are obtained. It is found that these solutions have the form of kink and antikink solitons, propagating on top of a continuous wave in the normal-and the anomalous-dispersion regime, respectively. The profile of the solutions is investigated in detail, and it is found that it depends on the relative importance of the nonlinearity and the dispersion on wave propagation. (C) 1997 Optical Society of America

Small-amplitude solitary structures for an extended nonlinear Schrodinger equation

A perturbative approach is used to obtain small-amplitude solitary structures for an extended nonlinear Schrodinger equation. These structures have the form of dark and anti-dark solitary wave solutions, closely connected with the Korteweg-deVries solitons. The solutions found are valid in wavelength regions, such as those applicable in the anomalous dispersion regime, which are not accessible by the conventional nonlinear Schrodinger equation. The dynamics of the derived structures in the presence of the Raman effect is also studied by means of a Korteweg-deVries-Burgers equation. The obtained results are applied to the problem of propagation of femtosecond duration pulses in nonlinear optical fibres

Multiscale expansions for a generalized cylindrical nonlinear Schrodinger equation

Considering a (3 + 1)-dimensional generalized nonlinear Schrodinger equation, we use the reductive multiscale expansion method to derive new evolution equations for small-amplitude solitary waves on a finite background. These equations are a combination of the so-called Johnson’s and a CI equation for the spatial solitons, and a CII equation for the temporal solitons. It is shown that the simplest one-dimensional soliton solutions to these two equations are either dark or anti-dark, depending on the type of the nonlinearity and a value of the background amplitude. It is also demonstrated that one can easily switch a dark soliton into an anti-dark one, increasing the background intensity. (C) 1999 Elsevier Science B.V. All rights reserved

SLOWLY VARYING FEMTOSECOND SOLITARY WAVES IN AXIALLY INHOMOGENEOUS OPTICAL FIBERS NEAR THE ZERO-DISPERSION POINT

A perturbed nonlinear Schrodinger equation that describes femtosecond pulse propagation in spatially (axially) inhomogeneous optical fibers near the zero-dispersion point is considered. This equation, which has varying coefficients, is analyzed by means of a multiple-scale perturbation technique. Approximate analytical results, valid up to the first order, concerning both the envelope function and the carrier wave number and frequency, are derived. Necessary conditions for envelope bright solitary-wave formation, as well as the solutions Typical results concerning the effect of the inhomogeneity on the solitary-wave propagation also are given. (C) 1995 Optical Society of Americ