Stability of Dipole Gap Solitons in Two-Dimensional Lattice Potentials

This chapter presents, for both self-focusing (SF) and self-defocusing (SDF) signs of the nonlinearity, the stability analysis for families of 2D gap solitons (GSs), in the first and second bandgaps, which are different from the "standard" extensively studied fundamental solitons and vortices supported by the SDF nonlinearity. The chapter introduces the model. It starts with the well-known 2D Gross-Pitaevskii/nonlinear Schrödinger equation for the mean-field complex wave function in the Bose-Einstein condensate (BEC). The chapter presents families of GSs residing in the first finite bandgap, under the SF nonlinearity. A stable subfamily of dipole solitons is identified. Bound states of the dipoles are also introduced and analyzed. A similar analysis is presented for dipole GSs in the second finite bandgap under the SDF nonlinearity. In addition, a stability region is found for a new GS family, which is sustained in the second bandgap under the SF nonlinearity.