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.
- P. Deift, X. Zhou, Long-time behavior of the non-focusing nonlinear Schro¨¨𝑜\ddot{o}over¨ start_ARG italic_o end_ARGdinger equation–a case study, Lectures in Mathematical Sciences, Graduate School of Mathematical Sciences, University of Tokyo, 1994.
- P. Deift, X. Zhou, Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Comm. Pure Appl. Math., 56(2003), 1029-1077.
- Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21(2009), 61-109.
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