An efficient Fourier spectral exponential time differencing method for the space-fractional nonlinear Schrödinger equations

A fourth-order time-discretization scheme based on the exponential time differencing approach with Fourier spectral method in space is proposed for the space-fractional nonlinear Schrödinger equations. The stability and convergence of the numerical scheme are discussed. It is shown that the proposed numerical scheme is fourth-order convergent in time and spectral convergent in space. Numerical experiments are performed on one-, two-, and three-dimensional fractional nonlinear Schrödinger equations and systems of two-, and three-dimensional equations. In addition, a realistic two-dimensional example with the solution of a singularity occurring in finite time is included. The results demonstrate accuracy, efficiency, and reliability of the scheme. Computational results arising from the experiments are compared with relevant known schemes, such as the fourth-order split-step Fourier method, the fourth-order explicit Runge-Kutta method, and a mass-conservative spectral method.