Numerical modeling of the KdV random wave field

The evolution of the initially random wave field with a Gaussian spectrum shape is studied numerically within the Korteweg–de Vries (KdV) equation. The properties of the KdV random wave field are analyzed: transition to a steady state, equilibrium spectra, statistical moments of a random wave field, and the distribution functions of the wave amplitudes. Numerical simulations are performed for different Ursell parameters and spectrum width. It is shown that the wave field relaxes to the stationary state (in statistical sense) with the almost uniform energy distribution in low frequency range (Rayleigh–Jeans spectrum). The wave field statistics differs from the Gaussian one. The growing of the positive skewness and non-monotonic behavior of the kurtosis with increase of the Ursell parameter are obtained. The probability of a large amplitude wave formation differs from the Rayleigh distribution.