Plane waves numerical stability of some explicit exponential methods for cubic Schrödinger equation

Abstract

Numerical stability when integrating plane waves of cubic Schr\"odinger equation is thoroughly analysed for some explicit exponential methods. We center on the following second-order methods: Strang splitting and Lawson method based on a one-parameter family of $2$-stage $2$nd-order explicit Runge-Kutta methods. Regions of stability are plotted and numerical results are shown which corroborate the theoretical results. Besides, a technique is suggested to avoid the possible numerical instabilities which do not correspond to continuous ones.