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We measure spectra, spatial correlation functions and probability densityfunctions (PDFs) for waves amplitudes for generalized one-dimensional nonlinearSchrodinger (NLS) equation of focusing type with saturated nonlinearity. Alladditional terms beyond the classical NLS equation are small. As initialconditions we use perturbed by weak noise modulationally unstable condensate.On the PDFs we observe power-law region for small and medium amplitudesfollowed by intermediate region and then Rayleigh far tail. Power-law regionappears starting from some critical levels of average amplitude and coefficientrelated to saturated nonlinearity, and then becomes more pronounced withaverage amplitude and saturation coefficient. Correlation of phases becomessignificant for large wave events and contributes about one order of magnitudeto the frequencies of their occurrence. Waves statistics for the consideredsystems turns out to be exceptionally stable against additional stochasticforces.

Waves statistics for generalized one-dimensional Nonlinear Schrodinger Equation with saturated nonlinearity