Distribution functions of the initiated KdV-like solitonic gas

The statistical properties of a sequence of spaced solitons and compactons (soliton gas) with random amplitudes and phases are studied based on the example of solitary waves – solutions of the generalized Korteweg-de Vries equation with power nonlinearity (including fractional nonlinearity). Such sequences are used to specify initial conditions in problems of modeling soliton turbulence. It is shown in the paper that in the case of a unipolar soliton gas, there is a critical density, so that the soliton gas is sufficiently rarefied regardless of the nonlinearity type in the generalized Korteweg-de Vries equation, which is associated with the repulsion of the same polarity solitons. On the contrary, the density of the bipolar soliton gas can be any, since solitons of different polarities are attracted. The first statistical moments of the wave field are calculated. The probability density functions of the soliton and compacton sequence are calculated. A feature in these functions in the region of small field values due to the overlap of exponential soliton tails is noted. This feature is absent in the field of compactons occupying the finite space volume. The paper provides calculation examples of distribution functions for various approximations to the distribution function of the solitary wave amplitudes.