Real-Valued Vector Modified Korteweg--de Vries Equation: Solitons Featuring Multiple Poles

TL;DR

This paper investigates the inverse scattering transform for a real-valued vector mKdV equation with multiple poles, developing a Riemann--Hilbert problem approach to construct multi-pole soliton solutions.

Contribution

It introduces a novel formulation using a matrix Riemann--Hilbert problem to handle multiple poles in multi-component systems of the vector mKdV equation.

Findings

Established existence and uniqueness of solutions for the formulated problem.

Reconstructed multi-pole soliton solutions in reflectionless cases.

Extended the inverse scattering framework to complex multi-pole scenarios.

Abstract

We delve into the inverse scattering transform of the real-valued vector modified Korteweg--de Vries equation, emphasizing the challenges posed by $N$ pairs of higher-order poles in the transmission coefficient and the enhanced spectral symmetry stemming from real-valued constraints. Utilizing the generalized vector cross product, we formulate an $(n+1)\times(n+1)$ matrix-valued Riemann--Hilbert problem to tackle the complexities inherent in multi-component systems. We subsequently demonstrate the existence and uniqueness of solutions for a singularity-free equivalent problem, adeptly handling the intricacies of multiple poles. In reflectionless cases, we reconstruct multi-pole soliton solutions through a system of linear algebraic equations.