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A. Zotov

Publications and source records attributed to A. Zotov.

At least 19 recordsLinked to original sources
arXiv2026

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

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.

nlin.SI↗
arXiv2026

Integrable open elliptic Toda chain with boundaries

In this letter we discuss the classical integrable elliptic Toda chain proposed by I. Krichever. Our goal is to construct an open elliptic Toda chain with boundary terms. This is achieved using the factorized form of the Lax matrix and gauge equivalence with the XYZ chain.

nlin.SI↗
arXiv2026

R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\rm BC}_n$ type

We consider the elliptic Calogero-Inozemtsev system of ${\rm BC}_n$ type with five arbitrary constants and propose $R$-matrix valued generalization for $2n\times 2n$ Takasaki's Lax pair. For this purpose we extend the Kirillov's ${\rm B}$-type associative Yang-Baxter equations to the similar relations depending on the spectral parameters and the Planck constants. General construction uses the elliptic Shibukawa-Ueno $R$-operator and the Komori-Hikami $K$-operators satisfying reflection equation. Then, using the Felder-Pasquier construction the answer for the Lax pair is also written in terms of the Baxter's 8-vertex $R$-matrix. As a by-product of the constructed Lax pair we also propose ${\rm BC}_n$ type generalization for the elliptic XYZ long-range spin chain, and we present arguments pointing to its integrability.

math-ph↗
arXiv2026

Classical elliptic ${\rm BC}_1$ Ruijsenaars-van Diejen model: relation to Zhukovsky-Volterra gyrostat and 1-site classical XYZ model with boundaries

We present a description of the classical elliptic ${\rm BC}_1$ Ruijsenaars-van Diejen model with 8 independent coupling constants through a pair of ${\rm BC}_1$ type classical Sklyanin algebras generated by the (classical) quadratic reflection equation with non-dynamical XYZ $r$-matrix. For this purpose, we consider the classical version of the $L$-operator for the Ruijsenaars-van Diejen model proposed by O. Chalykh. In ${\rm BC}_1$ case it is factorized to the product of two Lax matrices depending on 4 constants. Then we apply an IRF-Vertex type gauge transformation and obtain a product of the Lax matrices for the Zhukovsky-Volterra gyrostats. Each of them is described by the ${\rm BC}_1$ version of the classical Sklyanin algebra. In particular case, when 4 pairs of constants coincide, the ${\rm BC}_1$ Ruijsenaars-van Diejen model exactly coincides with the relativistic Zhukovsky-Volterra gyrostat. Explicit change of variables is obtained. We also consider another special case of the ${\rm BC}_1$ Ruijsenaars-van Diejen model with 7 independent constants. We show that it can be reproduced by considering the transfer matrix of the classical 1-site XYZ chain with boundaries. In the end of the paper, using another gauge transformation we represent the Chalykh's Lax matrix in a form depending on the Sklyanin's generators.

math-ph↗
arXiv2026

Large N limit of spectral duality in classical integrable systems

We describe the large $N$ limit of spectral duality between rational Gaudin models introduced by Adams, Harnad and Hurtubise. The limit of the ${\rm gl}_N$ model is performed by means of a noncommutative torus algebra represented by the fields on a torus with the Moyal-Weyl star product. We apply the approach developed by Hoppe, Olshanetsky and Theisen to the Gaudin-type models and describe the corresponding integrable field theory (2d hydrodynamics) on a torus. The dual model is the large $N$ limit of the ${\rm gl}_M$ Gaudin model with $N$ marked points written in the form of the Gaudin model with irregular singularities.

hep-th↗
arXiv2025

Classical r-matrix structure for elliptic Ruijsenaars chain and 1+1 field analogue of Ruijsenaars-Schneider model

The classical dynamical $r$-matrix structure for the periodic elliptic Ruijsenaars chain is described. The Poisson brackets for the monodromy matrix are calculated as well, thus providing Liouville integrability of the model. Next, we study its continuous non-relativistic limit and reproduce the Maillet type non-ultralocal $r$-matrix structure for the field analogue of the elliptic Calogero-Moser model.

math-ph↗
arXiv2025

Classical integrable spin chains of Landau-Lifshitz type from R-matrix identities

We describe a family of 1+1 classical integrable space-discrete models of the Landau-Lifshitz type through the usage of ansatz for $U$-$V$ (Lax) pair with spectral parameter satisfying the semi-discrete Zakharov-Shabat equation. The ansatz for $U$-$V$ pair is based on $R$-matrices satisfying the associative Yang-Baxter equation and certain additional properties. Equations of motion are obtained using a set of $R$-matrix identities. In the continuous limit we reproduce the previously known family of the higher rank Landau-Lifshitz equations.

nlin.SI↗
arXiv2025

Integrable deformations of principal chiral model from solutions of associative Yang-Baxter equation

We describe deformations of the classical principle chiral model and 1+1 Gaudin model related to ${\rm GL}_N$ Lie group. The deformations are generated by $R$-matrices satisfying the associative Yang-Baxter equation. Using the coefficients of the expansion for these $R$-matrices we derive equations of motion based on a certain ansatz for $U$-$V$ pair satisfying the Zakharov-Shabat equation. Another deformation comes from the twist function, which we identify with the cocentral charge in the affine Higgs bundle underlying the Hitchin approach to 2d integrable models.

math-ph↗
arXiv2025

Field analogue of the Ruijsenaars-Schneider model

We suggest a field extension of the classical elliptic Ruijsenaars-Schneider model. The model is defined in two different ways which lead to the same result. The first one is via the trace of a chain product of $L$-matrices which allows one to introduce the Hamiltonian of the model and to show that the model is gauge equivalent to a classical elliptic spin chain. In this way, one obtains a lattice field analogue of the Ruijsenaars-Schneider model with continuous time. The second method is based on investigation of general elliptic families of solutions to the 2D Toda equation. We derive equations of motion for their poles, which turn out to be difference equations in space with a lattice spacing $η$, together with a zero curvature representation for them. We also show that the equations of motion are Hamiltonian. The obtained system of equations can be naturally regarded as a field generalization of the Ruijsenaars-Schneider system. Its lattice version coincides with the model introduced via the first method. The limit $η\to 0$ is shown to give the field extension of the Calogero-Moser model known in the literature. The fully discrete version of this construction is also discussed.

math-ph↗
arXiv2025

Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model

We consider 1+1 field generalization of the elliptic Calogero-Moser model. It is shown that the Lax connection satisfies the classical non-ultralocal $r$-matrix structure of Maillet type. Next, we consider 1+1 field analogue of the spin Calogero-Moser model and its multipole (or multispin) extension. Finally, we discuss the field analogue of the classical IRF-Vertex correspondence, which relates utralocal and non-ultralocal $r$-matrix structures.

hep-th↗
arXiv2025

On the field analogue of elliptic spin Calogero-Moser model: Lax pair and equations of motion

The Lax pair for the field analogue of the classical spin elliptic Calogero-Moser is proposed. Namely, using the previously known Lax matrix we suggest an ansatz for the accompany matrix. The presented construction is valid when the matrix of spin variables ${\mathcal S}\in{\rm Mat}(N,\mathbb C)$ satisfies the condition ${\mathcal S}^2=c_0{\mathcal S}$ with some constant $c_0\in\mathbb C$. It is proved that the Lax pair satisfies the Zakharov-Shabat equation with unwanted term, thus providing equations of motion on the unreduced phase space. The unwanted term vanishes after additional reduction. In the special case ${\rm rank}(\mathcal S)=1$ we show that the reduction provides the Lax pair of the spinless field Calogero-Moser model obtained earlier by Akhmetshin, Krichever and Volvovski.

nlin.SI↗
arXiv2025

Interrelations between dualities in classical integrable systems and classical-classical version of quantum-classical duality

We describe the Ruijsenaars' action-angle duality in classical many-body integrable systems through the spectral duality transformation relating the classical spin chains and Gaudin models. For this purpose, the Lax matrices of many-body systems are represented in the multi-pole (Gaudin-like) form by introducing a fictitious spectral parameter. This form of Lax matrices is also interpreted as classical-classical version of quantum-classical duality.

math-ph↗
arXiv2024

Gauge equivalence of 1+1 Calogero-Moser-Sutherland field theory and higher rank trigonometric Landau-Lifshitz model

We consider the classical integrable 1+1 trigonometric ${\rm gl}_N$ Landau-Lifshitz models constructed by means of quantum $R$-matrices satisfying also the associative Yang-Baxter equation. It is shown that 1+1 field analogue of the trigonometric Calogero-Moser-Sutherland model is gauge equivalent to the Landau-Lifshitz model, which arises from the Antonov-Hasegawa-Zabrodin trigonometric non-standard $R$-matrix. The latter generalizes the Cherednik's 7-vertex $R$-matrix in ${\rm GL}_2$ case to the case of ${\rm GL}_N$. Explicit change of variables between the 1+1 models is obtained.

hep-th↗
arXiv2024

Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

We propose commuting sets of matrix-valued difference operators in terms of trigonometric ${\rm GL}(N|M)$-valued $R$-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric $R$-matrices are used for this purpose. The first is the one related to the affine quantized algebra ${\hat{\mathcal U}}_q({\rm gl}(N|M))$. The second is a graded version of the standard $\mathbb Z_n$-invariant $A_{n-1}$ type $R$-matrix. We show that being properly normalized the latter graded $R$-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the ${\rm gl}(N|M)$-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.

math-ph↗
arXiv2023

Gauge equivalence between 1+1 rational Calogero-Moser field theory and higher rank Landau-Lifshitz equation

In this paper we study 1+1 field generalization of the rational $N$-body Calogero-Moser model. We show that this model is gauge equivalent to some special higher rank matrix Landau-Lifshitz equation. The latter equation is described in terms of ${\rm GL}_N$ rational $R$-matrix, which turns into the 11-vertex $R$-matrix in the $N=2$ case. The rational $R$-matrix satisfies the associative Yang-Baxter equation, which underlies construction of the Lax pair for the Zakharov-Shabat equation. The field analogue of the IRF-Vertex transformation is proposed. It allows to compute explicit change of variables between the field Calogero-Moser model and the Landau-Lifshitz equation.

hep-th↗
arXiv2023

Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation

We study ${\rm GL}_N$ rational $R$-matrix, which turns into the 11-vertex $R$-matrix in the $N=2$ case. First, we describe its relations to dynamical and semi-dynamical $R$-matrices using the IRF-Vertex type transformations. As a by-product a new explicit form for ${\rm GL}_N$ $R$-matrix is derived. Next, we prove the quantum and the associative Yang-Baxter equations. A set of other $R$-matrix properties and $R$-matrix identities are proved as well.

math-ph↗
arXiv2022

Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\rm GL}_M$. In the scalar case $M=1$ these operators are the elliptic Macdonald-Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any $M$ is equivalent to a set of $R$-matrix identities. The proof of identities is based on the properties of elliptic $R$-matrix including the quantum and the associative Yang-Baxter equations. As an application of our results, we introduce elliptic generalization of q-deformed Haldane-Shastry model.

math.QA↗