Non-integrable soliton gas: The Schamel equation framework

TL;DR

This paper investigates the complex dynamics of non-integrable soliton gases using the Schamel equation, revealing statistical properties and deviations from integrable models like KdV and mKdV, especially in bipolar cases.

Contribution

It introduces a numerical study of soliton gases within the non-integrable Schamel equation framework, highlighting differences from classical integrable models.

Findings

Unipolar soliton gases align with KdV predictions.

Bipolar soliton gases show increased kurtosis and tail amplification.

Notable deviations from mKdV model in bipolar case.

Abstract

Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons interact elastically. However, soliton turbulence can also be a part of non-integrable dynamics, where long-lasting solutions in the form of almost solitons may exist. In the present paper, the complex dynamics of ensembles of solitary waves is studied within the Schamel equation using direct numerical simulations. Some important statistical characteristics (distribution functions, moments) are calculated numerically for unipolar and bipolar soliton gases. Comparison of results with integrable Korteweg-de Vries (KdV) and modified KdV (mKdV) models are given qualitatively. Our results agree well with the predictions of the KdV equation in the case of unipolar solitons. However, in the bipolar case, we observed a notable departure from the mKdV model, particularly in the behavior of kurtosis. The observed increase in kurtosis signifies the amplification of distribution function tails, which, in turn, corresponds to the presence of high-amplitude waves.