Nonlinear Space-fractional Schrodinger Equation Research Articles

A mass-energy preserving Galerkin FEM for the coupled nonlinear fractional Schrödinger equations

We consider the numerical simulation of the coupled nonlinear space fractional Schrodinger equations. Based on the Galerkin finite element method in space and the Crank-Nicolson (CN) difference method in time, a fully discrete scheme is constructed. Firstly, we focus on a rigorous analysis of conservation laws for the discrete system. The definitions of discrete mass and energy here correspond with the original ones in physics. Then, we prove that the fully discrete system is uniquely solvable. Moreover, we consider the unconditionally convergent properties (that is to say, we complete the error estimates without any mesh ratio restriction). We derive $L^{2}$ -norm error estimates for the nonlinear equations and $L^{\infty}$ -norm error estimates for the linear equations. Finally, some numerical experiments are included showing results in agreement with the theoretical predictions.

Galerkin-Legendre spectral schemes for nonlinear space fractional Schrödinger equation

In the paper, we first propose a Crank-Nicolson Galerkin-Legendre (CN-GL) spectral scheme for the one-dimensional nonlinear space fractional Schrodinger equation. Convergence with spectral accuracy is proved for the spectral approximation. Further, a Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional nonlinear space fractional Schrodinger equation is developed. The proposed schemes are shown to be efficient with second-order accuracy in time and spectral accuracy in space which are higher than some recently studied methods. Moreover, some numerical results are demonstrated to justify the theoretical analysis.