  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
$\mathcal{O}(t^{-1/2})$.
