Dynamical localization of gap-solitons by time periodic forces

The phenomenon of dynamical localization of matter wave solitons in optical lattices is first demonstrated and the conditions for its existence are discussed. In addition to the trapping linear periodic potential we use a periodic modulation of the nonlinearity in space to eliminate nonexistence regions of gap-solitons in reciprocal space. We show that when this condition is achieved, the observation of dynamical localization in true nonlinear regime becomes possible. The results apply to all systems described by the periodic nonlinear Schr\"odinger equation, including Bose-Einstein condensates of ultracold atoms trapped in optical lattices and arrays of waveguides or photonic crystals in nonlinear optics.