The Physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena

We review the different dynamical mechanisms leading to the emergence of coherent structures in physical systems described by the integrable one-dimensional nonlinear Schrödinger equation (1DNLSE) in the focusing regime. In this context, localized and coherent structures are very often associated to rogue wave events. We focus on one-dimensional optical experiments and in particular on (single mode) optical fibers experiments. In the focusing regime of 1DNLSE, the so-called modulation instability (MI), arising from nonlocal perturbation of the plane waves, is the most common phenomenon. Alongside the standard MI, other mechanisms are responsible for the emergence of rogue waves. We classify the different scenarii by considering those induced by small perturbations of unstable stationary state (the plane waves) and the ones arising from the self-focusing of large pulses without any perturbation. In the former case, the perturbations can be local, global, random or deterministic. In the latter case, the self-focusing dynamics can be observed both with isolated pulses or with large initial fluctuations of the optical power. We review the dynamics of emergence of localized structures in all these different scenarii.