Shock waves in discrete nonlinear Schrödinger equations

It is shown that in nonlinear differential–difference equations, which in the continuum limit are reduced to the nonlinear Schrödinger equation, localized excitations, reminding shock waves in liquids and gasses can propagate. Such waves may have either the conventional profile of shock waves or a shape of dark pulses evolving against a background. At the initial stages of evolution the shock waves are described by the equation ut+uux=0 and split out in a train of soliton-like pulses after the shock is developed.