On Cauchy problem to the modified Camassa-Holm equation: Painlev\'{e} asymptotics

We investigate the Painlev\'{e} asymptotics for the Cauchy problem of the modified Camassa-Holm (mCH) equation with decaying initial data \begin{align*}\nonumber &m_t+\left((u^2-u_x^2)m\right)_x+\kappa u_{x}=0, \ (x,t)\in\mathbb{R}\times\mathbb{R}^+,\\ &u(x,0)=u_0(x), \end{align*} where $u_0(x)\in H^{4,2}(\mathbb{R})$ and $\kappa$ is a constant. Recently, Yang and Fan (Adv. Math. 402 (2022) 108340) reported the long-time asymptotic results for the mCH equation in the different solitonic regions. The main purpose of our work is to study the asymptotic behavior of the mCH equation in the transition regions, which are the critical regions between the different solitonic regions. The key is to establish a connection between the solution for the Cauchy problem of the mCH equation in the transition region and the Painlev\'{e} II equation. With the $\bar{\partial}$-generalization of the Deift-Zhou nonlinear steepest descent method and double scaling limit technique, in two transition regions defined by \begin{align}\nonumber \mathcal{P}_{I}:=\{(x,t):0\leqslant \left|\frac{x}{t}-2\right|t^{2/3}\leqslant C\},~~~~\mathcal{P}_{II}:=\{(x,t):0\leqslant \left|\frac{x}{t}+1/4\right|t^{2/3}\leqslant C\}, \end{align} where $C>0$ is a constant, we obtain that the leading order approximation to the solution of the mCH equation can be expressed in terms of the Painlev\'{e} II equation.