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arXiv · 1911.07263

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

Anne Boutet de Monvel·Iryna Karpenko·Dmitry Shepelsky
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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.

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Anne Boutet de Monvel, Iryna Karpenko, Dmitry Shepelsky. 2019-12-03. A Riemann-Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions. https://doi.org/10.1063/1.5139519

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