Towards an analytic description of periodic anomalous waves in nature via the focusing NLS model.

Talk is based on joint work with P.G.Grinevich
The focusing NLS equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. We first show how the finite gap method adapts to the NLS Cauchy problem for a generic periodic initial perturbation of the unstable background solution, in the case of a finite number N of unstable modes, allowing one to construct the solution, to leading order, in terms of elementary functions of the initial data. In particular, if N=1, one obtains the analytic quantitative description of a Fermi-Pasta-Ulam recurrence of AWs already observed in real (water wave and nonlinear optics) and numerical experiments. Then we present the analytic description of the O(1) effect of a small loss or gain on the dynamics of periodic AWs, in full agreements with recent water wave and numerical experiments. If time remains, we shall discuss analogies and differences with AWs in other contexts: on the Ablowitz-Ladik lattice and in the massive Thirring model relativistic field theory (with F. Coppini).