Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain

Abstract

We discuss the classical elliptic Toda chain introduced by Krichever and the elliptic Ruijsenaars-Toda chain introduced by Adler, Shabat and Suris. It is shown that these models can be obtained as particular cases of the elliptic Ruijsenaars chain. We explain how the classical $r$-matrix structures are derived for these chains. Also, as a by-product, we prove that the elliptic Ruijsenaars-Toda chain is gauge equivalent to discrete Landau-Lifshitz model of XYZ type. The elliptic Toda chain is also gauge equivalent to XYZ chain with special values of the Casimir functions at each site.

29 pages, minor changes, details added to Section 5

References in corpus (13)

• Hitchin Systems - Symplectic Hecke Correspondence and Two-dimensional Version
• Vector bundles and Lax equations on algebraic curves
• Relativistic Classical Integrable Tops and Quantum R-matrices
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Field analogue of the Ruijsenaars-Schneider model
• Field theory generalizations of two-body Calogero-Moser models in the form of Landau-Lifshitz equations
• On factorized Lax pairs for classical many-body integrable systems
• On Elliptic Lax Systems on the Lattice and a Compound Theorem for Hyperdeterminants
• Discrete time Toda systems
• Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
• Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities
• Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model
• Classical elliptic ${\mathrm{BC}}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries