Two-component {CH} system: Inverse Scattering, Peakons and Geometry

TL;DR

This paper develops an inverse scattering transform method for the two-component Camassa-Holm (CH2) equation, enabling the explicit construction of multi-soliton solutions and exploring the equation's geometric properties.

Contribution

It introduces a Riemann-Hilbert problem approach for CH2 and constructs multi-soliton solutions, advancing the analytical understanding of this integrable system.

Findings

Formulation of an inverse scattering method for CH2

Explicit multi-soliton solutions for reflectionless potentials

Connection between CH2 solutions and geometric structures

Abstract

An inverse scattering transform method corresponding to a Riemann-Hilbert problem is formulated for CH2, the two-component generalization of the Camassa-Holm (CH) equation. As an illustration of the method, the multi - soliton solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data for CH2.