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

TL;DR

This paper extends the use of Hermite basis functions to simulate nonlinear Schrödinger equations without harmonic traps, introducing a new stable numerical method supported by theoretical and numerical evidence.

Contribution

It demonstrates the stability of Hermite functions for non-harmonic potentials and develops a novel unconditionally stable scheme for the derivative nonlinear Schrödinger equation.

Findings

Hermite basis functions are stable for non-harmonic Schrödinger equations.

A new unconditionally stable numerical method is proposed.

Numerical experiments confirm the theoretical stability and effectiveness.

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.