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

Abstract

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.