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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 solitonicregions \begin{align*} m_t+2\kappaq_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+\kappa,\\ q(x,0)=q_0(x)\inH^{4,2}(\mathbb R),~~x\in\mathbb R, ~~t>0, \end{align*} where $\kappa$ is apositive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problemcorresponding to the original problem is constructed to give the solution ofthe CH equation with the initial boundary value condition. Furthermore, bydeveloping the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepestdescent 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 asymptoticapproximations 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 twostationary points on the jump contour, and the asymptotic approximations can becharacterized with the soliton term confirmed by $N(j_0)$-soliton on discretespectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum withresidual error up to $O(t^{-1})$. Our results also confirm the solitonresolution conjecture for the CH equation with weighted Sobolev initial data inspace-time solitonic regions.

On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions