Additive splitting methods for the generalized nonlinear Schrödinger equation

Abstract

Splitting methods provide an efficient approach to solving evolutionary wave equations, especially in situations where dispersive and nonlinear effects on wave propagation can be separated, as in the generalized nonlinear Schrödinger equation (GNLSE). However, such methods are explicit and can lead to numerical instabilities. We study these instabilities in the context of the GNLSE. Results previously obtained for multiplicative splitting methods are extended to additive splittings. An easy-to-use estimate of the largest possible integration step is derived and confirmed by numerical experiments.