Inverse scattering transform of the coupled modified Korteweg-de Vries equation with nonzero boundary conditions

TL;DR

This paper extends the Riemann-Hilbert method to analyze the coupled modified Korteweg-de Vries equation with nonzero boundary conditions, revealing its solutions and dynamic behaviors through spectral analysis and soliton classification.

Contribution

It introduces a novel Riemann-Hilbert framework for the cmKdV equation with NZBCs, including spectral analysis, symmetry considerations, and soliton solutions classification.

Findings

Established the RH problem with four jump matrices.

Derived the asymptotic properties of eigenfunctions.

Classified soliton solutions based on discrete spectrum types.

Abstract

In this work, we extend the Riemann-Hilbert (RH) method in order to study the coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary conditions (NZBCs), and successfully find its solutions with their various dynamic propagation behaviors. In the process of spectral analysis, it is necessary to introduce Riemann surface to avoid the discussion of multi-valued functions, and to obtain the analytical and asymptotic properties needed to establish the RH problem. The eigenfunction have a column that is not analytic in a given region, so we introduce the auxiliary eigenfunction and the adjoint matrix, which is necessary to derive the analytical eigenfunctions. The eigenfunctions have three kinds of symmetry, which leads to three kinds of symmetry of the scattering matrix, and the discrete spectrum is also divided into three categories by us. The asymptoticity of the modified eigenfunction is derived. Based on the analysis, the RH problem with four jump matrices in a given area is established, and the relationship between the cmKdV equation and the solution of the RH problem is revealed. The residue condition of reflection coefficient with simple pole is established. According to the classification of discrete spectrum, we discuss the soliton solutions corresponding to three kinds of discrete spectrum classification and their propagation behaviors in detail.