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

TL;DR

This paper analyzes the long-time behavior of solutions to the Camassa-Holm equation in specific regions, using Riemann-Hilbert problems and the nonlinear steepest descent method to confirm the soliton resolution conjecture.

Contribution

It develops a detailed asymptotic analysis of the Camassa-Holm equation in space-time solitonic regions, employing a $ar{ ext{}}$-generalized steepest descent method, and confirms the soliton resolution conjecture.

Findings

Asymptotic expansions characterized by soliton and continuous spectrum contributions.

Different regions exhibit distinct stationary point structures affecting asymptotics.

Residual errors are quantified as $O(t^{-1+2\tau})$ and $O(t^{-1})$.

Abstract

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.