Modulational instability and spatial structures of the Ablowitz-Ladik equation

Spatial structures as a result of a modulational instability are obtained in the integrable discrete nonlinear Schrödinger equation (Ablowitz-Ladik equation). Discrete slow space variables are used in a general setting and the related finite differences are constructed. Analyzing the ensuing equation, we derive the modulational instability criterion from the discrete multiple scales approach. Numerical simulations in agreement with analytical studies lead to the disintegrations of the initial modulated waves into a train of pulses.