A Riemann-Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions

Abstract

The paper aims at developing the Riemann-Hilbert problem approach to the modified Camassa-Holm (mCH) equation in the case when the solution is assumed to approach a non-zero constant at the both infinities of the space variable. In this case, the spectral problem for the associated Lax pair equation has a continuous spectrum, which allows formulating the inverse spectral problem as a Riemann-Hilbert factorization problem with jump conditions across the real axis. We obtain a representation for the solution of the Cauchy problem for the mCH equation and also a description of certain soliton-type solutions, both regular and non-regular.

25 pages, no figures, we corrected item (ii) of the last theorem which was wrong

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Cited by in corpus (6)

• A view of the peakon world through the lens of approximation theory
• Transient asymptotics of the modified Camassa-Holm equation
• On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system
• On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$
• Generalized Heisenberg Ferromagnet type Equation and Modified Camassa-Holm Equation: Geometric Formulation, Soliton Solutions and Equivalence
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions