  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.
