Computing nonlinear Schrödinger equations with Hermite functions beyond harmonic traps
Provides a stable numerical method for nonlinear Schrödinger equations on unbounded domains, extending Hermite basis applicability beyond harmonic traps.
The paper shows that Hermite basis functions remain stable for Schrödinger equations without harmonic potential, enabling a novel unconditionally stable numerical method for the derivative nonlinear Schrödinger equation, supported by numerical examples.
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.