Dynamics of matter-wave solitons in a time-modulated two-dimensional   optical lattice
  By means of the variational approximation (VA) and systematic simulations, we study dynamics and stability boundaries for solitons in a two-dimensional (2D) self-attracting Bose-Einstein condensate (BEC), trapped in an optical lattice (OL) whose amplitude is subjected to the periodic time modulation (the modulation frequency, $\omega $, may be in the range of several KHz). Regions of stability of the solitons against the collapse and decay are identified in the space of the model's parameters. A noteworthy result is that the stability limit may reach the largest (100%) modulation depth, and the collapse threshold may exceed its classical value in the static lattice (which corresponds to the norm of Townes soliton). Minimum norm $% N_{\min}$ necessary for the stability of the solitons is identified too. It features a strong dependence on $\omega $ at a low frequencies, due to a resonant decay of the soliton. Predictions of the VA are reasonably close to results of the simulations. In particular, the VA helps to understand salient resonant features in the shape of the stability boundaries observed with the variation of $\omega $. 
Gennadiy Burlak, Boris A. Malomed