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We consider numerical instability that can be observed in simulations oflocalized solutions of the generalized nonlinear Schr\"odinger equation (NLS)by a split-step method where the linear part of the evolution is solved by afinite-difference discretization. Properties of such an instability cannot beinferred from the von Neumann analysis of the numerical scheme. Rather, theirexplanation requires tools of stability analysis of nonlinear waves, withnumerically unstable modes exhibiting novel features not reported for "real"unstable modes of nonlinear waves. For example, modes that cause numericalinstability of a standing soliton of the NLS are supported by the sides of thesoliton rather than by its core. Furthermore, we demonstrate that bothproperties and analyses of the numerical instability may be substantiallyaffected by specific details of the simulated solution; e.g., they aresubstantially different for standing and moving solitons of the NLS.

Instability of the finite-difference split-step method on the background of localized solutions of the generalized nonlinear Schr\"odinger equation