  We measure spectra, spatial correlation functions and probability density
functions (PDFs) for waves amplitudes for generalized one-dimensional nonlinear
Schrodinger (NLS) equation of focusing type with saturated nonlinearity. All
additional terms beyond the classical NLS equation are small. As initial
conditions we use perturbed by weak noise modulationally unstable condensate.
On the PDFs we observe power-law region for small and medium amplitudes
followed by intermediate region and then Rayleigh far tail. Power-law region
appears starting from some critical levels of average amplitude and coefficient
related to saturated nonlinearity, and then becomes more pronounced with
average amplitude and saturation coefficient. Correlation of phases becomes
significant for large wave events and contributes about one order of magnitude
to the frequencies of their occurrence. Waves statistics for the considered
systems turns out to be exceptionally stable against additional stochastic
forces.
