  In this work, we are devoted to study the Cauchy problem of the Camassa-Holm
(CH) equation with weighted Sobolev initial data in space-time solitonic
regions \begin{align*} m_t+2\kappa
q_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+\kappa,\\ q(x,0)=q_0(x)\in
H^{4,2}(\mathbb R),~~x\in\mathbb R, ~~t>0, \end{align*} where $\kappa$ is a
positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem
corresponding to the original problem is constructed to give the solution of
the CH equation with the initial boundary value condition. Furthermore, by
developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest
descent method, different long-time asymptotic expansions of the solution
$q(x,t)$ are derived. Four asymptotic regions are divided in this work: For
$\xi\in\left(-\infty,-\frac{1}{4}\right)\cup(2,\infty)$, the phase function
$\theta(z)$ has no stationary point on the jump contour, and the asymptotic
approximations can be characterized with the soliton term confirmed by
$N(j_0)$-soliton on discrete spectrum with residual error up to
$O(t^{-1+2\tau})$; For $\xi\in\left(-\frac{1}{4},0\right)$ and
$\xi\in\left(0,2\right)$, the phase function $\theta(z)$ has four and two
stationary points on the jump contour, and the asymptotic approximations can be
characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete
spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with
residual error up to $O(t^{-1})$. Our results also confirm the soliton
resolution conjecture for the CH equation with weighted Sobolev initial data in
space-time solitonic regions.
