标题： 关于修改的Camassa-Holm方程的柯西问题：Painlevé渐近性 显示英文标题

标题： On Cauchy problem to the modified Camassa-Holm equation: Painlevé asymptotics

摘要： 我们研究了具有衰减初始数据的修改Camassa-Holm（mCH）方程的柯西问题的Painlevé渐进行为\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*}其中$u_0(x)\in H^{4,2}(\mathbb{R})$和$\kappa$是一个常数。最近，Yang和Fan（Adv. Math. 402 (2022) 108340）报告了在不同的孤立子区域中mCH方程的长时间渐近结果。我们工作的主要目的是研究mCH方程在过渡区域中的渐近行为，这些过渡区域是不同孤立子区域之间的临界区域。关键是建立mCH方程柯西问题解在过渡区域与Painlevé II方程之间的联系。通过$\bar{\partial}$的Deift-Zhou非线性最速下降方法的推广和双尺度极限技术，在由\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}定义的两个过渡区域中，其中$C>0$是一个常数，我们得到mCH方程解的领先阶近似可以表示为Painlevé II方程的形式。 显示英文摘要

摘要： 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.