We consider several effects of the matter wave dynamics which can be observed in Bose-Einstein condensates embedded into optical lattices. For low-density condensates we derive approximate evolution equations, the form of which depends on relation among the main spatial scales of the system. Reduction of the Gross-Pitaevskii equation to a lattice model (the tight-binding approximation) is also presented. Within the framework of the obtained models we consider modulational instability of the condensate, solitary and periodic matter waves, paying special attention to different limits of the solutions, i.e. to smooth movable gap solitons and to strongly localized discrete modes. We also discuss how the Feshbach resonance, a linear force, and lattice defects affect the nonlinear matter waves. *

We discuss how to engineer the phase and amplitude of a complex order parameter using localized dissipative perturbations. Our results are applied to generate and control various types of atomic nonlinear matter waves (solitons) by means of localized dissipative defects. Introduction.-Dissipation is one of the main forces acting against the formation of nonlinear coherent structures in extended systems. When dissipation is present in systems without additional gain mechanisms, typically all excitations decay into the regime of linear waves. In this Letter we discuss how, contradicting this general principle, a localized dissipation can be used to engineer the phase of systems governed by complex order parameters through the generation of currents that imprint the required phases in the system. Although our results have general implications we discuss examples of systems ruled by an universal model of mathematical physics: the nonlinear Schrödinger equation (NLS). As to application fields we will focus on studying the possibility of controlling the phase of Bose-Einstein condensates (BECs) to generate different types of coherent structures.It was soon after the realization of BECs that their potential to support nonlinear coherent structures was recognized. [10]. Many techniques have been discussed to generate these structures but essentially all of them are "conservative" in nature based either on time-varying potentials, spatially selective optical transitions or on tailored interatomic interactions.In ultracold quantum gases dissipative mechanisms are related to inelastic collisions [11], interaction with the thermal component [12] or collapse dynamics [4,13]. While the latter can result in surviving matter-waves leading to a "non-destructive" effect of dissipation [14] on certain nonlinear structures, dissipation is generally found to damp the excitations and act against the generation or survival of coherent structures. In this Letter, however, we show that under certain conditions a properly localized dissipation, e.g. the one recently demonstrated with the help of a focused electron beam [15], can be used for generation and control of matter waves.

We consider localized modes (discrete breathers) of the discrete nonlinear Schrödin-ger equation i dψn dt = ψ n+1 + ψ n−1 − 2ψ n + σ|ψ n | 2 ψ n , σ = ±1, n ∈ Z. We study the diversity of the steady-state solutions of the form ψ n (t) = e iωt v n and the intervals of the frequency, ω, of their existence. The base for the analysis is provided by the anticontinuous limit (ω negative and large enough) where all the solutions can be coded by the sequences of three symbols "-", "0" and "+". Using dynamical systems approach we show that this coding is valid for ω < ω * ≈ −3.4533 and the point ω * is a point of accumulation of saddle-node bifurcations. Also we study other bifurcations of intrinsic localized modes which take place for ω > ω * and give the complete table of them for the solutions with codes consisting of less than four symbols.

We present a theory of dark soliton dynamics in trapped quasi-one-dimensional Bose-Einstein condensates, which is based on the local density approximation. The approach is applicable for arbitrary polynomial nonlinearities of the mean-field equation governing the system as well as to arbitrary polynomial traps. In particular, we derive a general formula for the frequency of the soliton oscillations in confining potentials. A special attention is dedicated to the study of the soliton dynamics in adiabatically varying traps. It is shown that the dependence of the amplitude of oscillations {\it vs} the trap frequency (strength) is given by the scaling law $X_0\propto\omega^{-\gamma}$ where the exponent $\gamma$ depends on the type of the two-body interactions, on the exponent of the polynomial confining potential, on the density of the condensate and on the initial soliton velocity. Analytical results obtained within the framework of the local density approximation are compared with the direct numerical simulations of the dynamics, showing remarkable match. Various limiting cases are addressed. In particular for the slow solitons we computed a general formula for the effective mass and for the frequency of oscillations.Comment: 16 pages, 4 figures. To appear in Phys. Rev.

We propose a method for generating shock waves in Bose-Einstein condensates by rapidly increasing the value of the nonlinear coefficient using Feshbach resonances. We show that in a cigar-shaped condensate there exist primary (transverse) and secondary (longitudinal) shock waves. We analyze how the shocks are generated in multidimensional scenarios and describe the related phenomenology.