Computing nonlinear Schrödinger equations with Hermite functions beyond harmonic traps

Hermite basis functions are a powerful tool for the spatial discretisation of Schrödinger equations with harmonic potential. In this work, we show that their stability properties extend to the simulation of Schrödinger 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 Schrödinger equation. Our theoretical results are supported with extensive numerical examples.