On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa-Holm equation

TL;DR

This paper constructs gauge copies of the AKNS model, reduces them to a two-component Camassa-Holm equation, and explores Bäcklund transformations linking solutions across different reductions, revealing new integrable structures.

Contribution

It introduces a novel gauge-based framework for deriving reductions of the AKNS hierarchy to the two-component Camassa-Holm equation and uncovers associated Bäcklund transformations.

Findings

Identified three classes of reductions based on the gauge angle 

Established Bäcklund transformations between solutions of different classes

Connected the AKNS hierarchy to a two-component Camassa-Holm model

Abstract

Different gauge copies of the Ablowitz-Kaup-Newell-Segur (AKNS) model labeled by an angle $\theta$ are constructed and then reduced to the two-component Camassa--Holm model. Only three different independent classes of reductions are encountered corresponding to the angle $\theta$ being 0, $\pi/2$ or taking any value in the interval $0<\theta<\pi/2$. This construction induces B\"{a}cklund transformations between solutions of the two-component Camassa--Holm model associated with different classes of reduction.