Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation
Résumé
Plane wave solutions to the cubic nonlinear Schrödinger 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, 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 that can both be verified if the time step-size is restricted by a CFL condition. The proof uses a Hamiltonian reduction and transformation and modulated Fourier expansions in time and provides detailed insight into the structure of the numerical solution.