标题： 基于黎曼-希尔伯特方法的两分量修正卡马斯-霍姆方程 显示英文标题

标题： A Riemann-Hilbert approach to the two-component modified Camassa-Holm equation

摘要： 在本文中，我们基于其Lax对，发展了一种Riemann-Hilbert（RH）方法来研究两组分修改的Camassa-Holm（2-mCH）方程的柯西问题。 进一步地，通过使用$\bar{\partial}$-广义的Deift-Zhou最陡下降方法对RH问题进行一系列变形，我们在四种不同空间时间区域中得到了2-mCH方程解的长时间渐近近似。 特别是我们引入了一种技术，将多跳跃矩阵分解统一成一种形式，这可以大大简化$\bar{\partial}$-最陡下降方法的计算。 显示英文摘要

摘要： In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax pair. Further via a series of deformations to the RH problem by using the $\bar{\partial}$-generalization of Deift-Zhou steepest descent method, we obtain the long-time asymptotic approximations to the solutions of the 2-mCH equation in four kinds of space-time regions. Especially we introduce a technique to unify multi-jump matrix factorizations into one form which can greatly simplify the calculation of the $\bar{\partial}$-steepest descent method.