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.
Categories: math.AP math-ph math.MP