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We analyze a numerical instability that occurs in the well-known split-stepFourier method on the background of a soliton. This instability is found to bevery sensitive to small changes of the parameters of both the numerical gridand the soliton, unlike the instability of most finite-difference schemes. % onthe background of a monochromatic wave, considered earlier in the literature.Moreover, the principle of ``frozen coefficients", in which variablecoefficients are treated as ``locally constant" for the purpose of stabilityanalysis, is strongly violated for the instability of the split-step method onthe soliton background. Our analysis explains all these features. It is enabledby the fact that the period of oscillations of the unstable Fourier modes ismuch smaller than the width of the soliton.

Stability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schr\"odinger equation