We present some recent results obtained in collaboration with P.M. Santini (Universiy of Roma I). We consider the periodic direct spectral problem for the self-focusing NLS equation in a special situation corresponding to a small perturbation of the constant solution. This model is actively used now as a model for generation of the rogue waves in nonlinear medias. We show that in this special situation all ingredients of the theta-functional formulas can be efficiently calculated as explicit power series with respect to the amplitude of the perturbation.

The results presented have been obtained in collaboration with P.M. Santini (University of Roma I). We consider the periodic direct spectral problem for the self-focusing NLS equation in a special situation corresponding to a small perturbation of the constant solution. This model is actively used now as a model for generation of the rogue waves in nonlinear medias. We show that in this special situation all ingredients of the theta-functional formulas can be efficiently calculated as explicit power series with respect to the amplitude of the perturbation.