Riemann surfaces close to degenerate ones in the theory of anomalous waves

One of the basic models for describing the generation of rogue waves due to modulation instability is the focusing Nonlinear Schrodinger equation with special Cauchy data; namely at the starting time one has a small perturbation of the unstable background. In the spatially-periodic setting the spectral curves corresponding to such initial data are small perturbations of the rational ones, and very simple explicit formulas can be obtained. For small number of unstable modes these formulas demonstrate a good agreement with the results of numerical simulations. The talk is based on the joint work with P. M. Santini.