Surface gap solitons in a nonlinear fractional Schrödinger equation.

We address the propagation dynamics of gap solitons at the interface between uniform media and an optical lattice in the framework of a nonlinear fractional Schrödinger equation. Different families of solitons residing in the first and second bandgaps of the Floquet-Bloch spectrum are revealed. They feature a combination of the unique properties of fractional diffraction effects, surface waves and gap solitons. The instability of solitons can be remarkably suppressed by the decrease of Lévy index, especially obvious for solitons in the second gaps. Additionally, we study the properties of multi-peaked solitons in fractional dimensions and find that they can be made completely stable in a wide region, provided that their power exceeds a critical value. Counterintuitively, at a small Lévy index, the instability region shrinks with the increase of the number of soliton peaks.