Dark and bright shock waves on oscillating backgrounds in a discrete nonlinear Schrödinger equation

Dark and bright shock waves on top of arbitrary oscillating backgrounds are investigated in a discrete version of the nonlinear Schr\"odinger equation. The existence of analytical curves in the parameter space corresponding to shock wave formation is established for arbitrary wave numbers k of the background radiation. The analysis is based on the small-amplitude approximation and is confirmed by direct numerical integrations of the system.