The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlev\'e asymptotics

TL;DR

This paper extends long-time and Painlevé asymptotics for the Camassa-Holm equation to solutions with initial data in a weighted Sobolev space, using a $ar{ ext{D}}$-generalized Deift-Zhou method and RH problem analysis.

Contribution

It introduces a new scale and analyzes the long-time behavior of solutions in different regions, providing detailed asymptotic expansions and error estimates for the CH equation.

Findings

Different asymptotic behaviors in four space-time regions.

Error estimates of order $t^{-1/2}$ and $t^{-3/4}$ for various regions.

Painlevé II equation describes transition regions.

Abstract

Based on the $\overline\partial$-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlev\'e asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space $ H^{4,2}(\mathbb{R})$. With a new scale $(y,t)$ and a RH problem associated with the initial value problem,we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane $\{ (y,t): -\infty <y<\infty, \ t> 0\}$ is divided into four asymptotic regions: 1. Fast decay region, $ y/t \in(-\infty,-1/4)$ with an error $\mathcal{O}(t^{-1/2})$; 2. Modulation-solitons region, $y/t \in(2,+\infty)$, the result can be characterized with an modulation-solitons with residual error $\mathcal{O}(t^{-1/2 })$; 3. Zakhrov-Manakov region,$y/t \in(0,2)$ and $y/t \in(-1/4,0)$. The asymptotic approximations is characterized by the dispersion term with residual error $\mathcal{O}(t^{-3/4})$; 4. Two transition regions, $|y/t|\approx 2$ and $|y/t| \approx -1/4$, the results are describe by the solution of Painlev\'e II equation with error order $\mathcal{O}(t^{-1/2})$.