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

TL;DR

This paper develops a Riemann-Hilbert framework to analyze the modified Camassa-Holm equation with non-zero boundary conditions, enabling explicit solution representations and soliton solution descriptions.

Contribution

It introduces a novel Riemann-Hilbert problem formulation for the mCH equation with non-zero boundaries, expanding analytical tools for this class of nonlinear equations.

Findings

Derived a Riemann-Hilbert problem for the mCH equation with non-zero boundary conditions.

Obtained explicit solution formulas for the Cauchy problem.

Described both regular and non-regular soliton solutions.

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.