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

TL;DR

This paper investigates the statistical properties and dynamics of multisoliton interactions in the Korteweg-de Vries equation, revealing different evolving patterns and quasi-stationary states in soliton gases with power-law amplitude distributions.

Contribution

It introduces a numerical method for constructing multisoliton solutions and analyzes their statistical behavior, highlighting the impact of amplitude distribution on collision patterns and moments.

Findings

Existence of qualitatively different multisoliton evolution patterns.

Statistical moments decrease during large soliton collisions.

Long intervals of quasi-stationary behavior in soliton gases with close amplitudes.

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.