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We study the existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices. Then, using these properties, we show that (time-dependent) “dynamical superlattices” can be used to controllably place, guide, and manipulate these solitons. In particular, we use numerical experiments to displace solitons by turning on a secondary lattice structure, transfer solitons from one location to another by shifting one superlattice substructure relative to the other, and implement solitonic “path-following”, in which a matter wave follows the time-dependent lattice substructure into oscillatory motion. (c) 2005 Elsevier B.V. All rights reserved.

We report the first observation of stationary necklace-like solitons. Such necklace structures were realized when a high-order vortex beam was launched appropriately into a two-dimensional optically induced photonic lattice. Our theoretical results obtained with continuous and discrete models show that the necklace solitons resulting from a charge-4 vortex have a pi phase difference between adjacent “pearls” and are formed in an octagon shape. Their stability region is identified.

We analyze the dynamics of dark matter-wave solitons on a Thomas-Fermi cloud described by the Gross-Pitaevskii equation with radial symmetry. One-dimensional, ring, and spherical dark solitons are considered, and the evolution of their amplitudes, velocities, and centers is investigated by means of a Lagrangian approach. In the case of large-amplitude oscillations, higher-order corrections to the corresponding equations of motion for the soliton characteristics are shown to be important in order to accurately describe its dynamics. The numerical results are found to be in very good agreement with the analytical predictions.

We investigate the dynamics of matter-wave solitons in the presence of a spatially varying atomic scattering length and nonlinearity. The dynamics of bright and dark solitary waves is studied using the corresponding Gross-Pitaevskii equation. The numerical results are shown to be in very good agreement with the predictions of the effective equations of motion derived by adiabatic perturbation theory. The spatially dependent nonlinearity is found to lead to a gravitational potential, as well as to a renormalization of the parabolic potential coefficient. This feature allows one to influence the motion of fundamental as well as higher-order solitons.

Motivated by the recent experimental achievements in the work with Bose-Einstein condensates (BECs), we consider bright matter-wave solitons, in the presence of a parabolic magnetic trap and a spatially periodic optical lattice (OL), in the attractive BEC. We examine pinned states of the soliton and their stability by means of perturbation theory. The analytical predictions are found to be in good agreement with numerical simulations. We then explore possibilities to use a time-modulated OL as a means of stopping and trapping a moving soliton, and of transferring an initially stationary soliton to a prescribed position by a moving OL. We also study the emission of radiation from the soliton moving across the combined magnetic trap and OL. We find that the soliton moves freely (without radiation) across a weak lattice, but suffers strong loss in deeper OLs.

We propose an experimentally tractable setting for observing an “instability” of a repulsive oscillating Bose-Einstein condensate that leads to the generation of dark solitons. We illustrate that when the trap of the condensate (which incorporates a localized impurity) is displaced so that the condensate flow is characterized by an atomic velocity larger than the local speed of sound, dark solitons are generated. The subcritical, near critical and supercritical are analyzed in detail. (c) 2005 Elsevier B.V. All rights reserved.

We consider the effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schrodinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation that predicts that broad quasicontinuum solitons are unstable, while their strongly anisotropic counterparts are stable. By means of numerical methods, it is found that, in the general case, the fundamental solitons and simplest on-site-centered vortex solitons (”vortex crosses”) feature enhanced or reduced stability areas, depending on the strength of the anisotropy. More surprising is the effect of anisotropy on the so-called “super-symmetric” intersite-centered vortices (”vortex squares”), with the topological charge S equal to the square’s size M: we predict in an analytical form by means of the Lyapunov-Schmidt theory, and confirm by numerical results, that arbitrarily weak anisotropy results in dramatic changes in the stability and dynamics in comparison with the degenerate, in this case, isotropic, limit.

An experimentally realizable scheme of periodic sign-changing modulation of the scattering length is proposed for Bose-Einstein condensates similar to dispersion-management schemes in fiber optics. Because of controlling the scattering length via the Feshbach resonance, the scheme is named Feshbach-resonance management. The modulational-instability analysis of the quasiuniform condensate driven by this scheme leads to an analog of the Kronig-Penney model. The ensuing stable localized structures are found. These include breathers, which oscillate between the Thomas-Fermi and Gaussian configuration, or may be similar to the 2-soliton state of the nonlinear Schrodinger equation, and a nearly static state (”odd soliton”) with a nested dark soliton. An overall phase diagram for breathers is constructed, and full stability of the odd solitons is numerically established.

We generalize recent work on parametric resonances for nonlinear Schr{\"o}dinger (NLS) type equations to the case of three dimensional Bose-Einstein condensates at zero temperatures. We show the possibility of such resonances in the three-dimensional case, using a moment method and numerical simulations., Comment: Accepted in Journal of Low Temperature Physics

We study soliton dynamics in a system of two linearly coupled discrete nonlinear Schrodinger equations, which describe the dynamics of a two-component Bose gas, coupled by an electromagnetic field, and confined in a strong optical lattice. When the nonlinear coupling strengths are equal, we use a unitary transformation to remove the linear coupling terms, and show that the existing soliton solutions oscillate from one species to the other. When the nonlinear coupling strengths are different, the soliton dynamics is numerically investigated and the findings are compared to the results of an effective two-mode model. The case of two linearly coupled Ablowitz-Ladik equations is also briefly discussed. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.