  We measure evolution of spectra, spatial correlation functions and
probability density functions (PDFs) of waves appearance for a set of
one-dimensional NLS-like equations of focusing type, namely for the classical
integrable Nonlinear Schrodinger equation (1), nonintegrable NLS equation
accounting for dumping (linear dissipation, two- and three-photon absorption)
and pumping terms (2) and generalized NLS equation accounting for six-wave
interactions, dumping and pumping terms (3). All additional terms beyond the
classical NLS equation are small. As initial conditions we choose seeded by
noise modulationally unstable solutions of the considered systems in the form
of (a) condensate for systems (1)-(3) and (b) cnoidal wave for the classical
NLS equation (1). We observe 'strange' results for the classical NLS equation
(1) with condensate initial condition including peak at zeroth harmonic in
averaged over ensemble spectra, non-decaying spacial correlation functions and
a 'breathing' region on the PDFs for medium waves amplitudes where frequency of
waves appearance oscillates with time, while the far-tails of the PDFs remain
Rayleigh ones. Addition of small dumping and pumping terms in model (2) breaks
integrability that results in absence of the peak at zeroth harmonic in
spectra, spacial correlation functions decaying to zero level and strictly
Rayleigh PDFs for waves amplitudes. For the classical NLS equation (1) with
cnoidal wave initial condition PDFs turn out to be significantly different from
Rayleigh ones with 'fat tails' in the region of large amplitudes where higher
waves appear more frequently, while for generalized NLS equation with six-wave
interactions, dumping and pumping terms (3) we demonstrate absence of
non-Rayleigh addition to the PDFs for zeroth six-wave interactions coefficient
and increase of non-Rayleigh addition with six-wave interactions term.
