The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics

Abstract

Based on the $\overline{\partial }$-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé 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 Painlevé II equation with error order $\mathcal{O}(t^{-1/2})$.

61 pages