Properties of synchronous collisions of solitons in the Korteweg - de Vries equation

Abstract

Synchronous collisions of solitons of the Korteweg -- de Vries equation are considered as a representative example of the interaction of a large number of solitons in a soliton gas. Statistical properties of the soliton field are examined for a model distribution of soliton amplitudes according to a power law. $N$-soliton solutions ($N \le 50$) are constructed with the help of a numerical procedure using the Darboux transformation and 100-digits arithmetic. It is shown that there exist qualitatively different patterns of evolving multisoliton solutions depending on the amplitude distribution. Collisions of a large number of solitons lead to the decrease of values of statistical moments (the orders from 3 to 7 have been considered). The statistical moments are shown to exhibit long intervals of quasi-stationary behavior in the case of a sufficiently large number of interacting solitons with close amplitudes. These intervals can be characterized by the maximum value of the soliton gas density and by ``smoothing'' of the wave fields in integral sense. The analytical estimates describing these degenerate states of interacting solitons are obtained.