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

TL;DR

This paper develops a Riemann-Hilbert framework to analyze the modified Camassa-Holm equation with step-like boundary conditions, enabling the characterization of solutions with non-zero asymptotic states.

Contribution

It introduces a novel Riemann-Hilbert approach tailored for the mCH equation with step-like boundaries, extending spectral analysis techniques to this setting.

Findings

Spectral functions characterized for initial data with step-like conditions

Representation of solutions via associated Riemann-Hilbert problem

Framework applicable to non-zero boundary asymptotics

Abstract

The paper aims at developing the Riemann-Hilbert (RH) approach for the modified Camassa-Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. We present detailed properties of spectral functions associated with the initial data for the Cauchy problem for the mCH equation and obtain a representation for the solution of this problem in terms of the solution of an associated RH problem.