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Plane wave solutions to the cubic nonlinear Schr\"odinger equation on a torushave recently been shown to behave orbitally stable. Under genericperturbations 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 powersof the smallness parameter. The present paper studies the question as towhether numerical discretizations by the split-step Fourier method inherit sucha generic long-time stability property. This can indeed be shown under acondition of linear stability and a non-resonance condition. They can both beverified if the time step-size is restricted by a CFL condition in the case ofa constant plane wave. The proof first uses a Hamiltonian reduction andtransformation and then modulated Fourier expansions in time. It providesdetailed insight into the structure of the numerical solution.

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