Darboux transformations, finite reduction groups and related Yang-Baxter maps

TL;DR

This paper constructs new Yang-Baxter maps using Darboux matrices invariant under finite reduction groups, revealing integrable structures related to NLS and DNLS equations and their vector generalizations.

Contribution

It introduces novel 6-dimensional Yang-Baxter maps derived from Darboux transformations for NLS and DNLS equations, including their reductions and vector extensions.

Findings

Constructed 6D YB maps for NLS and DNLS

Reduced to 4D YB maps on invariant leaves

Identified integrability and applications to symmetry-preserving maps

Abstract

In this paper we construct Yang-Baxter (YB) maps using Darboux matrices which are invariant under the action of finite reduction groups. We present 6-dimensional YB maps corresponding to Darboux transformations for the Nonlinear Schr\"odinger (NLS) equation and the derivative Nonlinear Schr\"odinger (DNLS) equation. These YB maps can be restricted to $4-$dimensional YB maps on invariant leaves. The former are completely integrable and they also have applications to a recent theory of maps preserving functions with symmetries \cite{Allan-Pavlos}. We give a $6-$ dimensional YB-map corresponding to the Darboux transformation for a deformation of the DNLS equation. We also consider vector generalisations of the YB maps corresponding to the NLS and DNLS equation.