Computing nonlinear Schr\"odinger equations with Hermite functions beyond harmonic traps

Abstract

Hermite basis functions are a powerful tool for the spatial discretisation of Schr\"odinger equations with harmonic potential. In this work, we show that their stability properties extend to the simulation of Schr\"odinger equations without harmonic potential, thus making them a natural basis for the computation of nonlinear dispersive equations on unbounded domains. Building on this spatial discretisation, we introduce a novel unconditionally stable numerical method for the derivative nonlinear Schr\"odinger equation. Our theoretical results are supported with extensive numerical examples.