Long-time asymptotic behavior of the modified Camassa-Holm equation

Yiling Yang, Engui Fan

Published 2021-01-07Version 1

In this paper, we apply $\overline\partial$ steepest descent method to study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation \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}, &u(x, 0)=u_{0}(x), \end{align} where $\kappa$ is a positive constant. It is shown that the solution of the Cauchy problem can be characterized via a the solution of Riemann-Hilbert problem. In a fixed space-time solitonic region $x/t\in(-\infty,-1/4-\delta_0]\cup[2+\delta_0,+\infty)$ with a positive small constant $\delta_0$, we further compute the long time asymptotic expansion of the solution $u(x,t)$, which implies soliton resolution conjecture and 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; the residual error order $\mathcal{O}(|t|^{-1+2\rho})$ from a $\overline\partial$ equation.