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We consider the implementation of the split-step method where the linear partof the nonlinear Schr\"odinger equation is solved using a finite-differencediscretization of the spatial derivative. The von Neumann analysis predictsthat this method is unconditionally stable on the background of aconstant-amplitude plane wave. However, simulations show that the method canbecome unstable on the background of a soliton. We present an analysisexplaining this instability. Both this analysis and the instability itself aresubstantially different from those of the Fourier split-step method, whichcomputes the spatial derivative by spectral discretization. We also found thatthe modes responsible for the numerical instability are supported by the sidesof the soliton, in contrast to unstable modes of linearized nonlinear waveequations, which (the modes) are supported by the soliton's core.

Instability of the finite-difference split-step method on the background of a soliton of the nonlinear Schrodinger equation