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

D. Murinov, A. Zotov

TL;DR

We study the elliptic Toda chain and the elliptic Ruijsenaars-Toda chain and show that each arises as a reduction of the periodic elliptic ${\rm GL}_N$ Ruijsenaars chain. We construct the classical $r$-matrix structures via center-of-mass reformulations and explicit Lax representations, and prove gauge equivalence to the classical XYZ chain (with appropriate Casimir constraints) and to a higher-rank Landau-Lifshitz XYZ system. We introduce an $\eta$-dependent formulation and a homogeneous/generalized $\eta_a$ scheme, and analyze the $\eta\to 0$ limit to obtain the elliptic Toda chain and its relation to XYZ. The work unifies these elliptic integrable systems by providing concrete Lax pairs, Poisson structures, and gauge-transforms linking them to spin-chain/LL-type models.

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.

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

TL;DR

We study the elliptic Toda chain and the elliptic Ruijsenaars-Toda chain and show that each arises as a reduction of the periodic elliptic Ruijsenaars chain. We construct the classical -matrix structures via center-of-mass reformulations and explicit Lax representations, and prove gauge equivalence to the classical XYZ chain (with appropriate Casimir constraints) and to a higher-rank Landau-Lifshitz XYZ system. We introduce an -dependent formulation and a homogeneous/generalized scheme, and analyze the limit to obtain the elliptic Toda chain and its relation to XYZ. The work unifies these elliptic integrable systems by providing concrete Lax pairs, Poisson structures, and gauge-transforms linking them to spin-chain/LL-type models.

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 -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.
Paper Structure (24 sections, 1 theorem, 214 equations)

This paper contains 24 sections, 1 theorem, 214 equations.

Table of Contents

  1. Introduction
  2. Elliptic ${\rm GL}_N$ Ruijsenaars chain
  3. Brief description of the model
  4. Classical $r$-matrix structure
  5. Coordinates in the center of mass frame
  6. Lax matrices and equations of motion
  7. Classical $r$-matrix structure
  8. Factorization and gauge equivalence with the higher rank Landau-Lifshitz XYZ chain
  9. Elliptic Ruijsenaars-Toda chain
  10. Hamiltonian and equations of motion
  11. Lax pair
  12. Classical $r$-matrix structure
  13. Gauge transformation to XYZ chain
  14. $\eta$-dependent description through relativistic top.
  15. Standard description of XYZ model.
  16. ...and 9 more sections

Key Result

Theorem 1

The Lax matrices $L^a(z)$ (w303) satisfy the following quadratic $r$-matrix structure: where and the matrices $s_{12}^a(z)$ are defined through Again, the delta-symbols $\delta^{a,\,b-1}$ and $\delta^{a,\,b+1}$ in (w320) are understood modulo $n$.

Theorems & Definitions (1)

  • Theorem 1