Collapse arrest in the space-fractional Schrödinger equation with an optical lattice* *Project supported by the National Natural Science Foundation of China (Grant No. 11947122), the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515110935), the Research Start-up Foundation of Dongguan University of Technology, the Guangdong Science and Technology Planning Program (Grant No. 2017A010102019), the Guangdong Province Natural Science Foundation

Abstract

The soliton solution and collapse arrest are investigated in the one-dimensional space-fractional Schrödinger equation with Kerr nonlinearity and optical lattice. The approximate analytical soliton solutions are obtained based on the variational approach, which provides reasonable accuracy. Linear-stability analysis shows that all the solitons are linearly stable. No collapses are found when the Lévy index 1 < α ≤ 2. For α = 1, the collapse is arrested by the lattice potential when the amplitude of perturbations is small enough. It is numerically proved that the energy criterion of collapse suppression in the two-dimensional traditional Schrödinger equation still holds in the one-dimensional fractional Schrödinger equation. The physical mechanism for collapse prohibition is also given.