On a conservative Fourier spectral Galerkin method for cubic nonlinear Schrödinger equation with fractional Laplacian

Abstract In this paper, a Crank–Nicolson Fourier spectral Galerkin method is proposed for solving the cubic fractional Schrodinger equation. Firstly, we discuss the mass and energy conservation laws for the nonlinear system and its corresponding fully discrete scheme. Secondly, the convergence with the spectral order accuracy in space and the second order of accuracy in time is exhibited. We perform one-dimensional calculation of the fractional derivative differential equation to verify our theoretical findings. Moreover, the proposed scheme is successfully applied to study two- and three-dimensional fractional quantum mechanics. Numerical results clearly exhibit that the fractional order can affect the shapes of soliton and rogue waves. The evolution of ground state solution can be clearly seen to be non-symmetrically configured when the fractional order becomes smaller.