The modified Camassa-Holm equation on the half line: a Riemann--Hilbert approach

2025/12/04 by Karpenko, Iryna, Shepelsky, Dmitry
#35Q15 #35Q55 #37K15 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.05274

Abstract

We consider the initial-boundary value (IBV) problem for the modified Camassa--Holm (mCH) equation m_t+(( u²- u_x²+2 u) m)_x = 0, m:= u- u_xx+1 on the half line x ≥ 0. We provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann--Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated with the initial and boundary values of the solution, whose compatibility is characterized in spectral terms.

Citations

• The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
• The modified Camassa-Holm equation in Lagrangian coordinates
• Liouville integrability of conservative peakons for a modified CH equation
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation

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