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Nuclear Phys. B, 2014, Volume 887, Pages 400–422 (Mi nphb8)  

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

A. Levinab, M. Olshanetskybc, A. Zotovcdb

a NRU HSE, Department of Mathematics, Myasnitskaya str. 20, Moscow, 101000, Russia
b ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218, Russia
c MIPT, Institutskii per. 9, Dolgoprudny, Moscow Region, 141700, Russia
d Steklov Mathematical Institute RAS, Gubkina str. 8, Moscow, 119991, Russia

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 rewrite 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 described.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-02-00594
14-01-00860
Ministry of Education and Science of the Russian Federation 11.G34.31.0023
Dynasty Foundation
Russian Academy of Sciences - Federal Agency for Scientific Organizations 19
The work was partially supported by RFBR grants 12-02-00594 (A.L. and M.O.) and 14-01-00860 (A.Z.). The work of A.L. was also partially supported by AG Laboratory GU-HSE, RF Government grant, ag. 11 11.G34.31.0023. The work of A.Z. was also partially supported by the D. Zimin's fund "Dynasty" and by the Program of RAS "Basic Problems of the Nonlinear Dynamics in Mathematical and Physical Sciences" Pi 19.


DOI: https://doi.org/10.1016/j.nuclphysb.2014.09.001


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Received: 16.06.2014
Revised: 21.08.2014
Accepted:01.09.2014
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