On the split-step method for the solution of nonlinear Schr"{o}dinger equation with the Riesz space fractional derivative

The aim of this paper is to extend the split-step idea for the solution of fractional partial differential equations. We consider the multidimensional nonlinear Schr"{o}dinger equation with the Riesz space fractional derivative and propose an efficient numerical algorithm to obtain it's approximate solutions. To this end, we first discretize the Riesz fractional derivative then apply the Crank-Nicolson and a split-step methods to obtain a numerical method for this equation. In the proposed method there is no need to solve the nonlinear system of algebraic equations and the method is convergent and unconditionally stable. The proposed method preserves the discrete mass which will be investigated numerically. Numerical results demonstrate the reliability, accuracy and efficiency of the proposed method.

Keywords:

finite difference method , Riesz space fractional derivatives , Unconditional stability , Schr {o}dinger equation