On dissipationless shock waves in a discrete nonlinear Schroedinger equation

It is shown that the generalized discrete nonlinear Schroedinger equation in a small amplitude approximation is reduced to a number of basic nonlinear integrable equations, such as the KdV, mKdV and KdV(2) equations, or to the fifth-order KdV equation, depending on the values of the parameters. In the dispersionless limit these equations lead to the wave-breaking phenomenon for general enough initial conditions, and, after taking into account small dispersion effects, result in the formation of dissipationless shock waves. The Whitham theory of modulations of nonlinear waves is used for an analytical description of such waves. Numerical simulations are used to obtain different types of bright and dark shocks