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

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.