Darboux transformations, finite reduction groups and related Yang-Baxter 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 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 \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.

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

References in corpus (1)

• Integrable Maps which Preserve Functions with Symmetries

Cited by in corpus (21)

• Grassmann extensions of Yang-Baxter maps
• Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere
• Reduction groups and related integrable difference systems of NLS type
• A noncommutative discrete potential KdV lift
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Darboux transformation with Dihedral reduction group
• Higher dimensional Automorphic Lie Algebras
• Anticommutative extension of the Adler map
• Poisson structures for lifts and periodic reductions of integrable lattice equations
• Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
• Entwining Yang-Baxter maps related to NLS type equations
• Relativistic Collisions as Yang-Baxter maps
• Automorphic Lie Algebras with dihedral symmetry
• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Birational solutions to the set-theoretical 4-simplex equation
• Two-component Yang-Baxter maps and star-triangle relations
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability
• Discrete integrable systems, Darboux transformations, and Yang-Baxter maps
• On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps