  Numerical simulations of the unidirectional random waves are performed within
the Korteweg -de Vries equation to investigate the statistical properties of
surface gravity waves in shallow water. Nonlinear evolution shows the
relaxation of the initial state to the quasi-stationary state differs from the
Gaussian distribution. Significant nonlinear effects lead to the asymmetry in
the wave field with bigger crests amplitudes and increasing of large wave
contribution to the total distribution, what gives the rise of the amplitude
probability, exceeded the Rayleigh distribution. The spectrum shifts in low
frequencies with the almost uniform distribution. The obtained results of the
nonlinear evolution of shallow-water waves are compared with known properties
of deep-water waves in the framework of the nonlinear Schrodinger equation.
