Dispersive shock waves: from water waves to nonlinear optics

Dispersive shock waves are strongly oscillating wave trains that spontaneously form and expand thanks to the action of weak dispersion, which contrasts the tendency, driven by the nonlinearity, to develop a gradient catastrophe. Here we review the basic concepts and recent progresses made in the description of such nonlinear waves, both in terms of experimental results and modelling. In particular, we discuss the formation of dispersive shocks in shallow water, which can be described in terms of Korteweg-de Vries orWhitham nonlocal equations.We contrast such results with those obtained in the field of nonlinear optics, described in terms of local or nonlocal nonlinear Schrödinger equations. Finally we show that a dispersive shock propagating under the action of small perturbations can radiate. A perturbative approach allows for the accurate prediction of the radiated frequencies.