We study spontaneous symmetry breaking in a system of two parallel quasi-one-dimensional traps (cores), equipped with optical lattices (OLs) and filled with a Bose-Einstein condensate (BEC). The cores are linearly coupled by tunneling (the model may also be interpreted in terms of spatial solitons in parallel planar optical waveguides with a periodic modulation of the refractive index). Analysis of the corresponding system of linearly coupled Gross-Pitaevskii equations (GPEs) reveals that spectral bandgaps of the single GPE split into subgaps. Symmetry breaking in two-component BEC solitons is studied in cases of the attractive (AA) and repulsive (RR) nonlinearity in both traps; the mixed situation, with repulsion in one trap and attraction in the other (RA), is considered too. In all the cases, stable asymmetric solitons are found, bifurcating from symmetric or antisymmetric ones (and destabilizing them), in the AA and RR systems, respectively. In either case, bi-stability is predicted, with a nonbifurcating stable branch, either antisymmetric or symmetric, coexisting with asymmetric ones. Solitons destabilized by the bifurcation tend to rearrange themselves into their stable asymmetric counterparts. In addition to the fundamental solitons, branches of twisted (odd) solitons in the AA system, and twisted bound states of fundamental solitons in both AA and RR systems, are found too. The impact of a phase mismatch, ∆, between the OLs in the two cores is also studied. It is concluded that ∆ = π/2 only mildly deforms the picture, while ∆ = π changes it drastically, replacing the symmetry-breaking bifurcations by pseudo-bifurcations, with the branch of asymmetric solutions asymptotically approaching its symmetric or antisymmetric counterpart (in the AA and RR system, respectively), rather than splitting off from it. Also considered is a related model, for a binary BEC in a single-core trap with the OL, assuming that the two species (representing different spin states of the same atom) are coupled by linear interconversion. In that case, the symmetry-breaking bifurcations in the AA and RR models switch their character, if the inter-species nonlinear interaction becomes stronger than the intra-species nonlinearity.

We introduce two-and one-dimensional (2D and 1D) models of a binary BEC (Bose-Einstein condensate) in a periodic potential, with repulsive interactions. We chiefly consider the most fundamental case of the inter-species repulsion with zero intra-species interactions. The same system may also model a mixture of two mutually repulsive fermionic species. Existence and stability regions for gap solitons (GSs) supported by the interplay of the inter-species repulsion and periodic potential are identified. Two-component GSs are constructed by means of the variational approximation (VA) and in a numerical form. The VA provides accurate description for the GS which is a bound state of two tightly-bound components, each essentially trapped in one cell of the periodic potential. GSs of this type dominate in the case of intra-gap solitons, with both components belonging to the first finite bandgap of the linear spectrum (only this type of solitons is possible in a weak lattice). Inter-gap solitons, with one component residing in the second bandgap, and intra-gap solitons which have both components in the second gap, are possible in a deeper periodic potential, with the strength essentially exceeding the recoil energy of the atoms. Inter-gap solitons are, typically, bound states of one tightly-and one loosely-bound components. In this case, results are obtained in a numerical form. The number of atoms in experimentally relevant situations is estimated to be ∼ 5, 000 in 2D intra-gap soliton, and ∼ 25, 000 in its inter-gap counterpart; in 1D solitons, it may be up to 10 5 . For 2D solitons, the stability is identified in direct simulations, while in the 1D case it is done via eigenfrequencies of small perturbations, and then verified by simulations. In the latter case, if the intra-gap soliton in the first bandgap is weakly unstable, it evolves into a stable breather, while unstable solitons of other types (in particular, inter-gap solitons) get completely destroyed. The intra-gap 2D solitons in the first bandgap are less robust, and in some cases they are completely destroyed by the instability. Addition of intra-species repulsion to the repulsion between the components leads to further stabilization of the GSs.

We consider a model of Bose-Einstein condensates, which combines a stationary optical lattice (OL) and periodic change of the sign of the scattering length (SL) due to the Feshbach-resonance management. Ordinary solitons and ones of the gap type being possible, respectively, in the model with constant negative and positive SL, an issue of interest is to find solitons alternating, in the case of the low-frequency modulation, between shapes of both types, across the zero-SL point. We find such alternate solitons and identify their stability regions in the 2D and 1D models. Three types of the dynamical regimes are distinguished: stable, unstable, and semi-stable. In the latter case, the soliton sheds off a conspicuous part of its initial norm before relaxing to a stable regime. In the 2D case, the threshold (minimum number of atoms) necessary for the existence of the alternate solitons is essentially higher than its counterparts for the ordinary and gap solitons in the static model. In the 1D model, the alternate solitons are also found only above a certain threshold, while the static 1D models have no threshold. In the 1D case, stable antisymmetric alternate solitons are found too. Additionally, we consider a possibility to apply the variational approximation (VA) to the description of stationary gap solitons, in the case of constant positive SL. It predicts the solitons in the first finite bandgap very accurately, and does it reasonably well in the second gap too. In higher bands, the VA predicts a border between tightly and loosely bound solitons.

Models of two-dimensional (2D) traps, with the double-well structure in the third direction, for Bose-Einstein condensate (BEC) are introduced, with attractive or repulsive interactions between atoms. The models are based on systems of linearly coupled 2D Gross-Pitaevskii equations (GPEs), where the coupling accounts for tunneling between the wells. Each well carries an optical lattice (OL) (stable 2D solitons cannot exist without OLs). The linear coupling splits each finite bandgap in the spectrum of the single-component model into two subgaps. The main subject of the work is spontaneous symmetry breaking (SSB) in two-component 2D solitons and localized vortices (SSB was not considered before in 2D settings). Using variational approximation (VA) and numerical methods, we demonstrate that, in the system with attraction or repulsion, SSB occurs in families of symmetric or antisymmetric solitons (or vortices), respectively. The corresponding bifurcation destabilizes the original solution branch and gives rise to a stable branch of asymmetric solitons or vortices. The VA provides for an accurate description of the emerging branch of asymmetric solitons. In the model with attraction, all stable branches eventually terminate due to the onset of collapse. Stable asymmetric solitons in higher finite bandgaps, and vortices with a multiple topological charge are found too. The models also give rise to first examples of embedded solitons and embedded vortices (the states located inside Bloch bands) in two dimensions. In the linearly-coupled system with opposite signs of the nonlinearity in the two cores, two distinct types of stable solitons and vortices are found, dominated by either the self-attractive component or the self-repulsive one. In the system with a mismatch between the two OLs, a pseudo-bifurcation is found: when the mismatch attains its largest value (π), the bifurcation does not happen, as branches of different solutions asymptotically approach each other but fail to merge.

We introduce a two-dimensional (2D) model which combines a checkerboard potential, alias the Kronig-Penney (KP) lattice, with the self-focusing cubic and self-defocusing quintic nonlinear terms. The beam-splitting mechanism and soliton multistability are explored in this setting, following the recently considered 1D version of the model. Families of single- and multi-peak solitons (in particular, five- and nine-peak species naturally emerge in the 2D setting) are found in the semi-infinite gap, with both branches of bistable families being robust against perturbations. For single-peak solitons, the variational approximation (VA) is developed, providing for a qualitatively correct description of the transition from monostability to the bistability. 2D solitons found in finite band gaps are unstable. Also constructed are two different species of stable vortex solitons, arranged as four-peak patterns ("oblique" and "straight" ones). Unlike them, compact "crater-shaped" vortices are unstable, transforming themselves into randomly walking fundamental beams.