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arXiv · 2402.13620

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

Kai Xu·Luman Ju·Engui Fan
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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.

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Kai Xu, Luman Ju, Engui Fan. 2024-02-21. A Riemann-Hilbert approach to the two-component modified Camassa-Holm equation. https://arxiv.org/abs/2402.13620

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