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

2012/05/31 by Sotiris Konstantinou-Rizos, Alexander Mikhailov · 3 citations
Mathematics · Physics and Astronomy · #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.1088/1751-8113/46/42/425201

18 pages, revised version. The format of the paper has changed, we added one section

arxiv created 2013/06/27 · arxiv updated 2015/06/05

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 Schrödinger (NLS) equation and the derivative Nonlinear Schrödinger (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 \citeAllan-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.

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