Distribution functions of gas of solitons of Korteweg – de Vries-type equation

The statistical properties of a rarefied soliton gas are studied using solitary waves – solutions of the generalized Korteweg – de Vries equation as an example. It is shown that there is a critical density of a soliton gas regardless of the type of nonlinearity in the generalized Korteweg – de Vries equation, which is associated with the repulsion of solitons of the same polarity. The first two statistical moments of the wave field (the mean value and the dispersion), which are simultaneously invariants of the Korteweg – de Vries-type equation, are calculated. The densities of the distribution function of a rarefied soliton gas are calculated. A feature in these functions in the region of small field values due to the overlap of the exponential tails of the solitons is noted.