Split-Step Methods for the Solution of the Nonlinear Schrödinger Equation

A split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrödinger equation. The space variable is discretized by means of a finite difference and a Fourier method. These methods are analyzed with respect to various physical properties represented in the NLS. In particular it is shown how a conservation law, dispersion and instability are reflected in the numerical schemes. Analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived. These results are demonstrated by numerical examples.