Classical integrable systems and soliton equations related to eleven-vertex R-matrix

Abstract

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\mathrm{gl}}_2$ rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to re-write the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of $n$-particle integrable systems with $2n$ constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau-Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is also described.

24 pages

References in corpus (3)

• Relativistic Classical Integrable Tops and Quantum R-matrices
• Vector bundles on degenerations of elliptic curves and Yang-Baxter equations
• Two body systems from sl(2,C)-tops

Cited by in corpus (10)

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• Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
• Higher rank 1+1 integrable Landau-Lifshitz field theories from associative Yang-Baxter equation
• 2d Integrable systems, 4d Chern-Simons theory and Affine Higgs bundles
• Relativistic elliptic matrix tops and finite Fourier transformations
• Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
• Gauge equivalence between 1+1 rational Calogero-Moser field theory and higher rank Landau-Lifshitz equation
• Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation
• Elliptic Ruijsenaars-Toda and elliptic Toda chains: classical r-matrix structure and relation to XYZ chain