Rigorous asymptotics of a KdV soliton gas

TL;DR

This paper provides a rigorous analysis of the long-term behavior of a broad class of KdV soliton gas solutions, revealing their asymptotic approach to cnoidal waves and zero, with descriptions in terms of elliptic functions.

Contribution

It introduces a rigorous asymptotic analysis of a new class of KdV solutions linked to a Riemann-Hilbert problem, connecting soliton gases to classical wave solutions.

Findings

Soliton gas approaches cnoidal waves as x→−∞

Soliton gas decays exponentially as x→+∞

Large-time asymptotics described by elliptic functions

Abstract

We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann--Hilbert problem which we show arises as the limit $N\to \infty$ of a gas of $N$-solitons. We show that this gas of solitons in the limit $N \to \infty$ is slowly approaching a cnoidal wave solution for $x \to - \infty$ (up to terms of order $\mathcal{O} (1/x)$), while approaching zero exponentially fast for $x\to+\infty$. We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.