### ?

## Hirota equation and Bethe ansatz in integrable models// Уравнение Хироты и анзац Бете в интегрируемых моделях

(Mathematical Sciences. 2013. No. 596. P. 7-12.

Translator: Takebe T.

We review the role of the Hirota equation and the tau-function in the theory of classical and quantum integrable systems.

Gorsky A., Zabrodin A., Zotov A., Journal of High Energy Physics 2014 No. 01 P. 070,28

In this paper we clarify the relationship between inhomogeneous quantum spin chains and classical integrable many-body systems. It provides an alternative (to the nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the spectrum of the quantum transfer matrix for the inhomogeneous glninvariant XXX spin chain on N sites with twisted ...

Added: July 15, 2014

Zabrodin A., / ИТЭФ. Series "ITEP-TH-17/12". 2012. No. 17.

The construction of the master T-operator recently suggested is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. At the same time it turns out to be the tau-function of an ...

Added: May 24, 2012

Zabrodin A., Proceedings of Physics and Mathematics of Nonlinear Phenomena 2014 Vol. 482 No. 012047 P. 10

This short note is a review of the intriguing connection between the quantum Gaudin model and the classical KP hierarchy recently established in [A.Alexandrov, S.Leurent, Z.Tsuboi, A.Zabrodin, The master T-operator for the Gaudin model and KP hierarchy, Nuclear Physics B 883 (2014) 173-223]. We construct the generating function of integrals of motion for the quantum ...

Added: July 15, 2014

Zabrodin A., Alexandrov A., Leurent S. et al., Nuclear Physics B 2014 Vol. 883 No. P. 173-223

Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we construct the master T-operator for the quantum Gaudin model with twisted boundary conditions and show that it satisfies the bilinear identity and Hirota equations for the classical KP ...

Added: July 15, 2014

Zabrodin A., Alexandrov A., Kazakov V. et al., Journal of High Energy Physics 2013 Vol. 09 P. 064

For an arbitrary generalized quantum integrable spin chain we introduce a “master T-operator” which represents a generating function for commuting quantum transfer matrices constructed by means of the fusion procedure in the auxiliary space. We show that the functional relations for the transfer matrices are equivalent to an infinite set of model-independent bilinear equations of ...

Added: November 14, 2013

Zabrodin A., Journal of Physics A: Mathematical and Theoretical 2013 Vol. 46 No. 18 P. 185203

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction. Similarly to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution ...

Added: April 29, 2013

Ogievetsky O., Pyatov P. N., Journal of Geometry and Physics 2021 Vol. 165 Article 104211

We establish the analogue of the Cayley–Hamilton theorem for the quantum matrix algebras of the symplectic type. We construct the algebra in which the quantum characteristic polynomial acquires a factorized form. The low-dimensional examples and the classical limit are discussed. ...

Added: March 18, 2021

Zabrodin A., Alexandrov A., Journal of Geometry and Physics 2013 Vol. 67 P. 37-80

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their ...

Added: February 16, 2013

Pyatov P. N., Ogievetsky O., / Cornell University. Series math "arxiv.org". 2020.

We establish the analogue of the Cayley--Hamilton theorem for the quantum matrix algebras of the symplectic type. ...

Added: January 26, 2021

Zabrodin A., Теоретическая и математическая физика 2013 Т. 174 № 1 С. 59-76

Недавно предложенная конструкция управляющего Т-оператора применяется к интегрируемым вершинным моделям и связанным с ними квантовым спиновым цепочкам с тригонометрическими R-матрицами. Управляющий Т-оператор - это производящая функция коммутирующих трансфер-матриц интегрируемых вершинных моделей, зависящая от бесконечного набора параметров.
В тоже время он оказывается тау-функцией интегрируемой иерархии классических солитонных уравнений в том смысле, что удовлетворяет тем же билинейным уравнения ...

Added: February 13, 2013

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2023 Article 033103

This work continues the study started in [1], where the exact densities of loops in the
O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions
were obtained. These densities are also equal to the densities of critical percolation clusters on the
forty five degree rotated square lattice rolled into a cylinder. ...

Added: February 1, 2024

Smirnov A., Matveenko S., Semenova E., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2015 Vol. 11

In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves. ...

Added: October 15, 2015

Pavel Pyatov, Povolotsky A. M., Rittenberg V., Journal of Statistical Mechanics: Theory and Experiment 2018 Vol. 2018 No. 053107 P. 1-26

We study the large deviation functions for two quantities characterizing the avalanche dynamics in the Raise and Peel model: the number of tiles removed by avalanches and the number of global avalanches extending through the whole system. To this end, we exploit their connection to the groundstate eigenvalue of the XXZ model with twisted boundary ...

Added: July 17, 2018

A A Trofimova, A M Povolotsky, Journal of Physics A: Mathematical and Theoretical 2022 Vol. 55 No. 2 Article 025202

We consider the particle current in the asymmetric avalanche process on a ring. It is known to exhibit a transition from the intermittent to continuous flow at the critical density of particles. The exact expressions for the first two scaled cumulants of the particle current are obtained in the large time limit t → ∞ ...

Added: August 15, 2022

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., SciPost Physic (Нидерланды) 2018 Vol. 4 No. 006 P. 1-30

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra U_q(gl_n). We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, ...

Added: September 13, 2018

Liashyk A., Slavnov N. A., Journal of High Energy Physics 2018 Vol. 06 No. 018 P. 1-31

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl_3-invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided ...

Added: September 13, 2018

Pogrebkov A., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2017 Vol. 13 No. 53 P. 1-14

Continuous symmetries of the Hirota difference equation, commuting with shifts of independent variables, are derived by means of the dressing procedure. Action of these symmetries on the dependent variables of the equation is presented. Commutativity of these symmetries enables interpretation of their parameters as “times” of the nonlinear integrable partial differential-difference and differential equations. Examples ...

Added: September 4, 2017

A.V.Zabrodin, Zotov A. V., Liashyk A. et al., Theoretical and Mathematical Physics 2017 Vol. 192 No. 2 P. 1141-1153

We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric(hyperbolic) functions. ...

Added: October 26, 2017

А. М. Поволоцкий, Физика элементарных частиц и атомного ядра 2021 Т. 52 № 2 С. 459-529

This review gives a survey of some results about systems of interacting particles and
the laws characterizing their behavior on large scales, which are common for a number of
phenomena uniˇed under the notion of the KardarÄParisiÄZhang universality class. ...

Added: August 15, 2022

Feigin B. L., Jimbo M., Mukhin E., Journal of Physics A: Mathematical and Theoretical 2017 Vol. 50 No. 46 Article 464001

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors.
That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model.
We also discuss the (gl(m),gl(n)) duality of XXZ models in ...

Added: November 5, 2020

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., Russian Mathematical Surveys 2017 Vol. 72 No. 1 P. 33-99

Bethe vectors are found for quantum integrable models associated with the supersymmetric Yangians in terms of the current generators of the Yangian double . The method of projections onto intersections of different types of Borel subalgebras of this infinite-dimensional algebra is used to construct the Bethe vectors. Calculation of these projections makes it possible to express the ...

Added: October 26, 2017

Povolotsky A. M., Приезжев В. Д., Теоретическая и математическая физика 2011 Т. 169 № 1 С. 167-175

Изучаются корреляционные функции простого полностью асимметричного процесса с исключающим взаимодействием в дискретном времени с обратной последовательной динамикой. Доказывается детерминантная формула для обобщенной функции Грина, которая описывает переходы между положениями частиц в заданные моменты времени. В качестве примера вычисляется корреляционная функция токов, т. е. совместное распределение вероятностей времен, необходимых каждой частице, чтобы пройти данное расстояние. Асимптотический ...

Added: February 27, 2013

Povolotsky A. M., Journal of Statistical Mechanics: Theory and Experiment 2019 No. 074003 P. 1-22

We establish the exact laws of large numbers for two time additive quantities in the raise and peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the ...

Added: October 8, 2019

Pyatov P. N., de Gier J., Zinn-Justin P., Journal of Combinatorial Theory, Series A 2009 Vol. 116 P. 772-794

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik–Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley–Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ 2-weighted ...

Added: October 16, 2012