Stability analysis of the split-step Fourier method on the background of a soliton of the nonlinear Schrödinger equation
For researchers using split-step Fourier methods for nonlinear Schrödinger equations, this work identifies and explains a previously uncharacterized instability on soliton backgrounds, which is more sensitive than instabilities on monochromatic waves.
The paper analyzes a numerical instability in the split-step Fourier method when applied on the background of a soliton of the nonlinear Schrödinger equation, showing it is highly sensitive to parameter changes and violates the frozen coefficients principle. The analysis explains these features by leveraging the timescale separation between unstable mode oscillations and soliton width.
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. % on the background of a monochromatic wave, considered earlier in the literature. 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 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.