Propagation of fundamental gap solitons in the fractional Schrödinger equation with combined linear and nonlinear optical lattices

The propagation of two-dimensional fundamental gap solitons is investigated in the framework of fractional Schrödinger equation with combined linear and nonlinear optical lattices. It is revealed that solitons exist in the first gap and can be stable in a large region. With a phase tilt, the fundamental gap solitons undergo oscillation during the propagation. And the oscillation is enhanced with the increase of the tilt. In addition, propagation of fundamental gap solitons are also studied by longitudinal modulation of Lévy index, nonlinear strength, and period of nonlinear lattices, respectively. With the sudden variation of Lévy index, the fundamental gap solitons undergo oscillation and become unstable, whereas the gradual variation generates the stable soliton. However, both sudden and gradual decreases of nonlinear strength can lead to the distortion of the soliton due to the weak nonlinearity. Interestingly, the small longitudinal modulation of the period of the nonlinear lattice has little effect on the propagation of fundamental gap soliton, while the effect of the large variation of the period of the nonlinear lattice is apparent.