Large-space and large-time asymptotics for the mKdV soliton gas with any odd genus

TL;DR

This paper analyzes the large-space and large-time asymptotic behavior of the mKdV soliton gas of any odd genus, providing explicit descriptions using Riemann-theta functions and the nonlinear steepest descent method.

Contribution

It offers a comprehensive asymptotic analysis of the mKdV soliton gas for any odd genus, including explicit formulas and regional descriptions.

Findings

Asymptotics expressed with Riemann-theta functions of genus 2n-1

Global large-time asymptotic description with uniform error estimates

Division of the half-plane into 2n+1 regions with distinct asymptotics

Abstract

We study the large-space and large-time asymptotic behavior of the soliton gas of genus $2n-1$ for the mKdV equation with $n\in \mathbb{N}_+$. As $x \to +\infty$, we show that the large-space asymptotics of the mKdV soliton gas can be expressed with the Riemann-theta function of genus $2n-1$. For large $t$, based on the nonlinear steepest descent method and $g$-function approach, we establish a global large-time asymptotic description of the mKdV soliton gas. The half-plane $\{(x,t):-\infty<x<+\infty, t>0\}$ is divided into $2n+1$ separated regions. In each region, the large-time asymptotics of the mKdV soliton gas is given by using the Riemann-theta functions and uniform error estimation.