Almost degenerate Riemann surfaces in the theory of anomalous (rogue) waves

Now the theory of anomalous (rogue) waves in nonlinear systems is a topic of great interest from both theoretical and applied point of view. The modern theory of rogue waves essentially uses soliton equations including focusing Nonlinear Schrödinger equation. Spatially-periodic solutions of soliton equation can be constructed using the finite-gap (algebro-geometrical) approach. But generic finite-gap formulas are not very convenient for practical applications and require additional effectivization.
Using the focusing Nonlinear Schrödinger equation and the focusing Davey-Stewardson 2 equations as main examples, we demonstrate that special solutions describing the generation of anomalous waves due to modulation instability correspond to spectral curves close to rational ones. For such curves the finite-gap solutions can be approximated be elementary functions (different for different time regions), and al parameters can be explicitly expressed in terms of the Cauchy data. (Joint work with P.M. Santini)