Three-dimensional Bose–Einstein gap solitons in optical lattices with fractional diffraction

Compared with low-dimensional solitons that are widely studied in various realizable nonlinear physical systems, the properties and dynamics of three-dimensional solitons and vortices have not been well disclosed yet. Using numerical simulations and theoretical analysis, we here address the existence, structural property, and dynamics of three-dimensional gap solitons and vortices (with topological charge s=1) of Bose–Einstein condensates moving by Lévy flights (characterized by fractional diffraction operators, Lévy index 1<α≤2) in optical lattices. We stress that previously the localized modes have only been revealed in low-dimensional nonlinear fractional systems in one- and two-dimensional periodic potentials, our study presented here thus drives the associated nonlinear-wave research into three-dimensional configurations. The three-dimensional optical lattices exhibit a nontrivial wide band-gap feature, within which the matter-wave localized gap modes could be excited. The stability and instability regions of both three-dimensional gap modes are obtained via direct perturbed simulations, shedding light on multidimensional soliton physics in nonlinear fractional systems with periodic potentials.