Plane wave stability of the split-step Fourier method for the nonlinear Schr\"odinger equation

TL;DR

This paper investigates whether the split-step Fourier method preserves the long-time stability of plane wave solutions in the nonlinear Schrödinger equation, demonstrating conditions under which numerical stability aligns with analytical stability.

Contribution

It establishes that the split-step Fourier method inherits long-time stability of plane waves under specific linear stability and non-resonance conditions, verified by a CFL restriction.

Findings

Stability is preserved under a CFL condition for constant plane waves.

Hamiltonian reduction and modulated Fourier expansions provide detailed structure analysis.

Numerical stability aligns with analytical stability under certain conditions.

Abstract

Plane wave solutions to the cubic nonlinear Schr\"odinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a non-resonance condition. They can both be verified if the time step-size is restricted by a CFL condition in the case of a constant plane wave. The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.