  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.
