On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions

TL;DR

This paper analyzes the long-term behavior of solutions to the modified Camassa-Holm equation in different solitonic regions, deriving asymptotic expansions and confirming soliton stability using advanced spectral and Riemann-Hilbert techniques.

Contribution

It introduces a novel asymptotic analysis of the mCH equation in various space-time regions, utilizing the $ar{ ext{D}}$ steepest descent method to characterize soliton interactions.

Findings

Asymptotic solutions are characterized by modulated N-solitons with localized interactions.

Residual errors decay at rates $igO(|t|^{-1+2 ho})$ and $igO(|t|^{-3/4})$ in different regions.

Results confirm the soliton resolution conjecture and stability of N-soliton solutions.

Abstract

In this paper, we study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation in the solitonic region \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+\kappa u_{x}=0, \quad m=u-u_{x x}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $\kappa$ is a positive constant. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a Riemann-Hilbert (RH) problem. Further using the $\overline\partial$ generalization of Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(x,t)$ in different space-time solitonic region of $x/t$. These asymptotic approximations can be characterized with an $N(\Lambda)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region with diverse residual error order from $\overline\partial$ equation: $\mathcal{O}(|t|^{-1+2\rho})$ for $\xi=\frac{y}{t}\in(-\infty,-0.25)\cup(2,+\infty)$ and $\mathcal{O}(|t|^{-3/4})$ for $\xi=\frac{y}{t}\in(-0.25,2)$. Our results also confirm the soliton resolution conjecture and asymptotically stability of N-soliton solutions for the mCH equation.