Instability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation

We analyze a numerical instability that occurs in the well-known split-step Fourier method on the background of a soliton. This instability is found to be very sensitive to small changes of the parameters of both the numerical grid and the soliton, unlike the instability of most finite-difference schemes. Moreover, the principle of “frozen coefficients”, in which variable coefficients are treated as “locally constant” for the purpose of stability analysis, is strongly violated for the instability of the split-step method on the soliton background. Our analysis quantitatively explains all these features. It is enabled by the fact that the period of oscillations of the unstable Fourier modes is much smaller than the width of the soliton. Our analysis is different from the von Neumann analysis in that it requires spatially growing or decaying harmonics (not localized near the boundaries) as opposed to purely oscillatory ones.