Statistical properties of extreme soliton collisions

TL;DR

This paper investigates the statistical behavior of large-scale soliton collisions, revealing a quasi-stationary state with reduced moments and potential analytical estimates, primarily in the context of the Korteweg--de Vries equation.

Contribution

It introduces a statistical framework for analyzing soliton collisions and characterizes a quasi-stationary state with reduced statistical moments, applicable to a broad class of soliton-supporting equations.

Findings

Wave field becomes smoother during same-sign soliton collisions.

Statistical moments freeze at high soliton numbers and slow amplitude decay.

Analytical estimates of high-order moments are possible in the small-dispersion limit.

Abstract

Synchronous collisions between a large number of solitons are considered in the context of a statistical description. It is shown that during the interaction of solitons of the same signs the wave field is effectively smoothed out. When the number of solitons increases and the sequence of their amplitudes decay slower, the focused wave becomes even smoother and the statistical moments get frozen for a long time. This quasi-stationary state is characterized by greatly reduced statistical moments and by the density of solitons close to some critical value. This state may be treated as the small-dispersion limit, what makes it possible to analytically estimate all high-order statistical moments. While the focus of the study is made on the Korteweg--de Vries equation and its modified version, a much broader applicability of the results to equations that support soliton-type solutions is discussed.