Asymptotics of step-like solutions for the Camassa-Holm equation

TL;DR

This paper analyzes the long-time behavior of solutions to the Camassa-Holm equation with step-like initial data, revealing complex asymptotic structures involving modulated finite-gap functions and solitons.

Contribution

It introduces a detailed asymptotic analysis of the Camassa-Holm equation using the nonlinear steepest descent method and $g$-function approach, uncovering rich solution structures.

Findings

Identification of distinct asymptotic regions in the $x,t$-plane

Description of solutions as sums of modulated finite-gap functions and solitons

Application of the nonlinear steepest descent method to integrable PDEs

Abstract

We study the long-time asymptotics of solution of the Cauchy problem for the Camassa-Holm equation with a step-like initial datum. By using the nonlinear steepest descent method and the so-called $g$-function approach, we show that the Camassa-Holm equation exhibits a rich structure of sharply separated regions in the $x,t$-half-plane with qualitatively different asymptotics, which can be described in terms of a sum of modulated finite-gap hyperelliptic or elliptic functions and a finite number of solitons.