A self-consistent theory of a cylindrically shaped Bose-Einstein condensate (BEC) periodically modulated by a laser beam is presented. We show, both analytically and numerically, that modulational instability/stability is the mechanism by which wave functions of soliton type can be generated in a cylindrically shaped BEC subject to a one-dimensional optical lattice. The theory explains why bright solitons can exist in a BEC with positive scattering length and why condensates with negative scattering length can be stable and give rise to dark solitary pulses

The existence of stable solitons in two-and three-dimensional (2D and Typeset using REVT E X 1

We show that the phenomenon of modulational instability in arrays of Bose-Einstein condensates confined to optical lattices gives rise to coherent spatial structures of localized excitations. These excitations represent thin disks in 1D, narrow tubes in 2D, and small hollows in 3D arrays, filled in with condensed atoms of much greater density compared to surrounding array sites. Aspects of the developed pattern depend on the initial distribution function of the condensate over the optical lattice, corresponding to particular points of the Brillouin zone. The long-time behavior of the spatial structures emerging due to modulational instability is characterized by the periodic recurrence to the initial low-density state in a finite optical lattice. We propose a simple way to retain the localized spatial structures with high atomic concentration, which may be of interest for applications. Theoretical model, based on the multiple scale expansion, describes the basic features of the phenomenon. Results of numerical simulations confirm the analytical predictions.

In the present Letter we use the Wannier function basis to construct lattice approximations of the nonlinear Schrödinger equation with a periodic potential. We show that the nonlinear Schrödinger equation with a periodic potential is equivalent to a vector lattice with long-range interactions. For the case-example of the cosine potential we study the validity of the so-called tight-binding approximation i.e., the approximation when nearest neighbor interactions are dominant. The results are relevant to Bose-Einstein condensate theory as well as to other physical systems like, for example, electromagnetic wave propagation in nonlinear photonic crystals. PACS numbers: 42.50. Ar,42.81.Dp Interplay between nonlinearity and periodicity is the focus of numerous recent studies in different branches of modern physics. The theory of Bose-Einstein condensates (BEC) within the framework of the mean field approximation [1] is one of them. Recent interest in the effects of periodicity in BEC's has been stimulated by a series of remarkable experiments realized with BEC's placed in a potential created by a laser field [2] (the socalled optical lattice). Nonlinearity and periodicity have been observed to introduce fundamental changes in the properties of the system. On the one hand periodicity modifies the spectrum of the underlying linear system resulting in the potential of existence of new coherent structures, which could not exist in a homogeneous nonlinear system. On the other hand, nonlinearity renders accumulation and transmission of energy possible in "linearly" forbidden frequency domains; this, in turn, results in field localization. This situation is fairly general and can be found in other applications, such as the theory of electromagnetic wave propagation in periodic media (so-called photonic crystals) [3].The study of nonlinear evolution equations with periodic coefficients is a challenging and interdisciplinary problem. This problem cannot be solved exactly in the general case and thus gives rise to various approximate approaches. One of them, borrowed from the theory of solid state [4], is the reduction of a continuous evolution problem to a lattice problem (i.e., reduction of a partial differential equation to a differential-difference one). It turns out that the relation between the properties of periodic and discrete problems is indeed rather deep (for a recent discussion of the relevant connections see e.g., [5] and references therein). Following the solid state terminology here we will refer to a discrete approximation when only nearest neighbor interactions are taken into account as a tight-binding model. This model has recently been employed in the description of BEC in an optical lattice [6]. One of the advantages of the lattice approach is that it allows one to obtain strongly localized configurations, the so-called intrinsic localized modes (ILM) (also called breathers) [7], in a rather simple way. These entities correspond to gap solitons of the original continuum model. In the above mentioned wo...

Using the variational approximation(VA) and direct simulations, we find stable 2D and 3D solitons in the self-attractive Gross-Pitaevskii equation (GPE) with a potential which is uniform in one direction ($z$) and periodic in the others (but the quasi-1D potentials cannot stabilize 3D solitons). The family of solitons includes single- and multi-peaked ones. The results apply to Bose-Einstein condensates (BECs) in optical lattices (OLs), and to spatial or spatiotemporal solitons in layered optical media. This is the first prediction of {\em mobile} 2D and 3D solitons in BECs, as they keep mobility along $z$. Head-on collisions of in-phase solitons lead to their fusion into a collapsing pulse. Solitons colliding in adjacent OL-induced channels may form a bound state (BS), which then relaxes to a stable asymmetric form. An initially unstable soliton splits into a three-soliton BS. Localized states in the self-repulsive GPE with the low-dimensional OL are found too.Comment: 4 pages, 5 figure