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arXiv · 1205.4910

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

Sotiris Konstantinou-Rizos·Alexander Mikhailov
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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.

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Sotiris Konstantinou-Rizos, Alexander Mikhailov. 2012-05-22. Darboux transformations, finite reduction groups and related Yang-Baxter maps. https://doi.org/10.1088/1751-8113/46/42/425201

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