Bäcklund Transformations as exact integrable time-discretizations for the trigonometric Gaudin model

Abstract

We construct a two-parameter family of Bäcklund transformations for the trigonometric classical Gaudin magnet. The approach follows closely the one introduced by E.Sklyanin and V.Kuznetsov (1998,1999) in a number of seminal papers, and takes advantage of the intimate relation between the trigonometric and the rational case. As in the paper by A.Hone, V.Kuznetsov and one of the authors (O.R.) (2001) the Bäcklund transformations are presented as explicit symplectic maps, starting from their Lax representation. The (expected) connection with the XXZ Heisenberg chain is established and the rational case is recovered in a suitable limit. It is shown how to obtain a "physical" transformation mapping real variables into real variables. The interpolating Hamiltonian flow is derived and some numerical iterations of the map are presented.

17 pages, 2 figures

References in corpus (17)

• Separation of Variables. New Trends.
• New integrable systems related to the relativistic Toda lattice
• Few remarks on Baecklund transformations for many-body systems
• Backlund transformations for finite-dimensional integrable systems: a geometric approach
• What is the relativistic Volterra lattice?
• Dynamical R-Matrices for Integrable Maps
• Quantum and Classical Integrable Systems
• Separation of Variables in BC-type Gaudin Magnet
• Separation of variables and Bäcklund transformations for the symmetric Lagrange top
• Gaudin Models and Bending Flows: a Geometrical Point of View
• Backlund transformations for the sl(2) Gaudin magnet
• Discrete Dubrovin Equations and Separation of Variables for Discrete Systems
• Integrable discretizations of the spin Ruijsenaars-Schneider models
• Algebraic extensions of Gaudin models
• Bäcklund Transformations for the Trigonometric Gaudin Magnet
• Toda-Darboux maps and vertex operators
• Bäcklund transformations for the rational Lagrange chain

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