Computer Science – Numerical Analysis
Scientific paper
2010-08-29
Computer Science
Numerical Analysis
28 pages, 5 figures This is the original manuscript submitted to the journal Numerical methods for PDEs. A revised version, in
Scientific paper
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.
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