A conservative spectral Galerkin method for the coupled nonlinear space-fractional Schrödinger equations

In this paper, a conservative spectral Galerkin method, which is based on the Crank-Nicolson method for the temporal discretization and Legendre spectral Galerkin method for the spatial discretization, is proposed to solve the coupled nonlinear space-fractional Schrödinger equations. We proved that the proposed method satisfies the mass and energy conservation laws in the discrete sense. Moreover, a rigorous analysis of the unique solvability and optimal error estimate in the L² -norm of the Crank-Nicolson spectral Galerkin method are derived. In order to compute the nonlinear system efficiently, we introduce a linear iterative algorithm in implementation. A series of numerical experiments are carried out to illustrate the efficiency of the method.