Computing nonlinear Schrödinger equations with Hermite functions beyond harmonic traps

Abstract

Hermite basis functions are a classical tool for the spatial discretisation of Schrödinger equations with harmonic potential. In this work, we prove that their favourable stability properties extend to Schrödinger equations without a trap: the free Schrödinger flow is stable in the weighted Sobolev spaces ${\Sigma }^k$ which govern the convergence of Hermite spectral methods. This makes the Hermite basis a natural discretisation for a larger class of nonlinear Schrödinger equations posed on the full space ${\mathbb{R}}^d$, avoiding artificial periodisation and the associated distortion of the dynamics incurred by domain truncation in Fourier methods. Within this framework we provide a rigorous fully discrete convergence analysis of a splitting method for the cubic nonlinear Schrödinger equation. In addition, by combining the Hermite basis with a gauge transform, we introduce a novel, fully explicit, unconditionally stable numerical method for the derivative nonlinear Schrödinger equation. Our theoretical results are supported with numerical examples across various nonlinearities and dimensions, which showcase the accuracy, stability and resulting efficiency of this Hermite basis approach.