We examine one-and two-dimensional (1D and 2D) models of linearly coupled lattices of the discrete-nonlinear-Schrödinger type. Analyzing ground states of the systems with equal powers in the two components, we find a symmetry-breaking phenomenon beyond a critical value of the squared l 2 -norm. Asymmetric states, with unequal powers in their components, emerge through a subcritical pitchfork bifurcation, which, for very weakly coupled lattices, changes into a supercritical one. We identify the stability of various solution branches. Dynamical manifestations of the symmetry breaking are studied by simulating the evolution of the unstable branches. The results present the first example of spontaneous symmetry breaking in 2D lattice solitons. This feature has no counterpart in the continuum limit, because of the collapse instability in the latter case.

Dated: To appear in Phys. Rev. A)Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with nr = 0, 1 and 2 radial nodes respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge m, i.e., vorticity. It is found that for repulsive interatomic interactions only the ground state is linearly stable throughout the parameter range examined. Furthermore, this is true for topological charges m = 0 or m = 1; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing/attractive case the ground state with nr = 0 and m = 0 can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.

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We examine the dynamics of a bright solitary wave in the presence of a repulsive or attractive localized "impurity" in Bose-Einstein condensates (BECs). We study the generation and stability of a pair of steady states in the vicinity of the impurity as the impurity strength is varied. These two new steady states, one stable and one unstable, disappear through a saddle-node bifurcation as the strength of the impurity is decreased. The dynamics of the soliton is also examined in all the cases (including cases where the soliton is offset from one of the relevant fixed points). The numerical results are corroborated by theoretical calculations which are in very good agreement with the numerical findings.

Motivated by the recent experimental achievements in the work with Bose-Einstein condensates ͑BECs͒, we consider bright matter-wave solitons, in the presence of a parabolic magnetic trap and a spatially periodic optical lattice ͑OL͒, 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 radiation͒ across a weak lattice, but suffers strong loss in deeper OLs.