Integrability in action: Solitons, instability and rogue waves

Integrable nonlinear equations modeling wave phenomena play an important role in understanding and predicting experimental observations. Indeed, even if approximate, they can capture important nonlinear effects because they can be derived, as amplitude modulation equations, by multiscale perturbation methods from various kind of wave equations, not necessarily integrable, under the assumption of weak dispersion and nonlinearity. Thanks to the mathematical property of being integrable, a number of powerful computational techniques is available to analytically construct special interesting solutions, describing coherent structures such as solitons and rogue waves, or to investigate patterns as those due to shock waves or behaviors caused by instability. This chapter illustrates a selection of these techniques, using first the ubiquitous Nonlinear Schrödinger (NLS) equation as a prototype integrable model, and moving then to the Vector Nonlinear Schrödinger (VNLS) equation as a natural extension to wave coupling.