Shelf solutions and dispersive shocks in a discrete NLS equation: Effects of nonlocality

Abstract

We study shelf-like breathers and dispersive shock phenomena in a discrete nonlinear Schrödinger (DNLS) equation with a nonlocal nonlinearity. The system models laser light propagation in waveguide arrays made from a nematic liquid crystal substratum. Shelf-like breathers are studied in the regime of small linear intersite coupling, and we report some new theoretical existence and stability results. We also study numerically the evolution from nearby dam-break and more general jump initial conditions for stronger linear intersite coupling. In the defocusing case, we see rarefaction and shock wave profiles, superposed with oscillations. Some of the hyperbolic features of the observed profiles are described approximately by continuous NLS hydrodynamics. Nonlocality is seen to lead to some smoothing of the rapid oscillations seen in the local DNLS.