Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

TL;DR

This paper investigates the complex interaction between a dense soliton gas and a trial soliton in the mKdV equation, revealing detailed analytical structures and deriving the soliton's phase shift and oscillatory velocity behavior.

Contribution

It provides a novel analytical framework using Fredholm determinants and Riemann-Hilbert problems to analyze soliton-gas interactions and derive the soliton's phase shift and velocity oscillations.

Findings

Explicit expression of the solution via Fredholm determinants and Riemann-Hilbert problem.

Derivation of the local phase shift of the gas after soliton passage.

Identification of oscillatory velocity behavior of the soliton peak.

Abstract

We analyze the case of a dense mKdV soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann-Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.