Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere

TL;DR

This paper develops a novel Darboux transformation method for the vector sine-Gordon equation, leading to new integrable systems, Bäcklund transformations, and a vector Yang-Baxter map on a sphere.

Contribution

It introduces a new Darboux transformation approach for Lax operators with reduction groups, deriving related integrable equations and maps.

Findings

Derived Bäcklund transformations for the vector sine-Gordon equation.

Established a new Lax operator associated with the Darboux transformation.

Constructed a vector Yang-Baxter map and integrable discrete sine-Gordon equation.

Abstract

We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its B\"acklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of B\"acklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations we derive new vector Yang-Baxter map and integrable discrete vector sine-Gordon equation on a sphere.