On the long-time asymptotic of the modified Camassa-Holm equation with nonzero boundary conditions in space-time solitonic regions

TL;DR

This paper analyzes the long-time behavior of solutions to the modified Camassa-Holm equation with nonzero boundary conditions, revealing soliton resolution and stability in different space-time regions using spectral and steepest descent methods.

Contribution

It provides a detailed asymptotic analysis of the mCH equation with nonzero boundary conditions, confirming the soliton resolution conjecture and demonstrating asymptotic stability.

Findings

Solution characterized by N-soliton and error in certain regions

Soliton resolution holds with discrete spectrum and continuous spectrum contributions

Soliton solutions are asymptotically stable

Abstract

We investigate the long-time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with nonzero boundary conditions in different regions \begin{align*} &m_{t}+\left((u^2-u_x^2)m\right)_{x}=0,~~ m=u-u_{xx}, ~~ (x,t)\in\mathbb{R}\times\mathbb{R}^{+},\\ &u(x,0)=u_{0}(x),~~\lim_{x\to\pm\infty} u_{0}(x)=1,~~u_{0}(x)-1\in H^{4,1}(\mathbb{R}), \end{align*} where $m(x,t=0):=m_{0}(x)$ and $m_{0}(x)-1\in H^{2,1}(\mathbb{R})$. Through spectral analysis, the initial value problem of the mCH equation is transformed into a matrix RH problem on a new plane $(y,t)$, and then using the $\overline{\partial}$-nonlinear steepest descent method, we analyze the different asymptotic behaviors of the four regions divided by the interval of $\xi=y/t$ on plane $\{(y,t)|y\in(-\infty,+\infty), t>0\}$. There is no steady-state phase point corresponding to the regions $\xi\in(-\infty,-1/4)\cup(2,\infty)$. We prove that the solution of mCH equation is characterized by $N$-soliton solution and error on these two regions. In $\xi\in(-1/4,0)$ and $\xi\in(0,2)$, the phase function $\theta(z)$ has eight and four steady-state phase points, respectively. We prove that the soliton resolution conjecture holds, that is, the solution of the mCH equation can be expressed as the soliton solution on the discrete spectrum, the leading term on the continuous spectrum, and the residual error. Our results also show that soliton solutions of the mCH equation with nonzero condition boundary are asymptotically stable.