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