# Statistical Equilibrium of the Korteweg-de Vries and Benjamin-Ono Unidirectional Soliton Models

## TL;DR

This paper investigates the statistical equilibrium of unidirectional soliton models described by Korteweg-de Vries and Benjamin-Ono equations, addressing the challenge of describing thermal equilibrium in a dilute soliton gas.

## Contribution

It introduces a stochastic source approach to characterize the equilibrium distribution of soliton momenta in these models.

## Key findings

- Derived the equilibrium distribution of soliton momenta.
- Demonstrated the role of stochastic sources in soliton thermalization.
- Provided a framework for understanding soliton gas thermodynamics.

## Abstract

The Korteweg-de Vries and Benjamin-Ono nonlinear wave equations can describe solitary waves, all of which propagate in the same direction and which emerge from collisions with their shapes unchanged. There are technical challenges to giving a description of the thermal equilibrium of a dilute gas of such solitons. These challenges can be overcome by considering a stochastic source of new solitons located at $x=0$. The total intensity and the momentum distributions of the solitons emitted by such sources determine the equilibrium distribution of soliton momenta for $x>0$.