Dispersive Shock Waves: From Water Waves to Nonlinear Optics

Abstract

Rogue and dispersive shock waves are phenomena that occur in nonlinear dispersive media. They have been studied in different fields of physics, including ocean waves, nonlinear optics, Bose-Einstein condensates, plasma physics, etc. They are apparently very different phenomena; however, they are both characterised by the development of extremes: while the amplitude of a rogue wave reaches large values, shock waves develop extreme gradients. The presence of rogue and shock waves in an incoherent wave system drastically influences its statistical properties.During the last 15 years, the field of "rogue waves" has experienced a very quick development. The original motivation of the research was related mostly to the occasional measurement of extreme waves on the surface of the ocean and to the occurrence of various accidents caused by the impact of large amplitude waves on ships. Only more recently, it has been shown that extreme light fluctuations could be observed in an optical fibre; the research activity has now broadened up to the creation of a new field in its own.A rogue wave is the manifestation of a process of focussing of energy. One of the most accredited explanations of the formation of rogue waves (at least in an idealised case) is the modulational instability process by which a small perturbation of a plane wave can grow exponentially fast in time. This mechanism has been known from the late sixties, but only at the beginning of the new century, it has been associated to rogue waves. The nonlinear stages of the modulational instability are described by exact solutions of the nonlinear Schrödinger equation. Those solutions, named breathers, have been considered as the prototypes of rogue waves in the ocean.The development of an infinite gradient in finite time (gradient catastrophe) with consequent wave-breaking is probably a less baleful phenomenon, nonetheless of extreme nature. Mathematically, a classical shock wave is a discontinuous (weak) solution describing propagation beyond a breaking point where an infinite derivative develops. Starting from the last century, in several branches of physics, it has been recognised that the dissipation plays an important role in regularising the jump leading to physical shock waves characterised by strong but finite gradients. A completely different non-trivial dynamics may result when dispersion dominates. v vi Preface