The focusing nonlinear Schrödinger equation is one of the main models for describing the generation of anomalous waves (also known as killer waves) due to modulation instability. In this case, the Cauchy problem with special initial conditions is investigated - at the zero moment of time we have a small perturbation of the spatially constant solution.

The nonlinear Schrödinger equation is a completely integrable system, and its periodic solutions are constructed in terms of the Riemann theta functions, but it is quite difficult to directly use theta functional formulas. We have shown that, due to the presence of a small parameter in the initial perturbation, the spectral curves turn out to be almost degenerate, and the solutions for the general perturbation are approximated with high accuracy by elementary functions (different for different time intervals), and all parameters of the approximating solutions are explicitly calculated using the Fourier coefficients of the initial perturbation. These results have already been used in optical experiments of two groups. In conclusion, we give formulas describing the effect of low friction on the frequency of anomalous waves for the simplest non-trivial case of one unstable mode.