Alternate solitons:: Nonlinearly managed one- and two-dimensional solitons in optical lattices

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.