Periodic rogue waves and perturbation theory

Rogue waves (RWs) are extreme waves of anomalously large amplitude with respect to the surrounding waves, arising apparently from nowhere and disappearing without leaving any trace. The simplest basic model describing the generation of RWs, due to modulation instability in nonlinear media, is the self-focusing Nonlinear Schr\"odinger (NLS) equation. Here we study the problem in the spatially periodic setting, using the finite gap method, for initial perturbations of the unstable background solution, in the case of a finite number of unstable modes. In addition, we construct a perturbation theory allowing one to study the effects of a small perturbation of the NLS equation on the dynamics of periodic rogue waves