r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R^2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.     

同一个非动力学r矩阵所对应的有理Ruijsenaars–Schneider模型(n=2)及Calogero–Moser模型

构造了有理Ruijsenaars–Schneider模型(n=2)的新的Lax算子,发现相应的r矩阵是非动力学的,它与Calogero–Moser模型具有相同的r矩阵.     

一个经典完全可积的非共形约束的SL(2,R)Wess-Zumino-Novikov-Witten模型(Ⅰ)

通过在ＳＬ（２，Ｒ）Ｗｅｓｓ－Ｚｕｍｉｎｏ－Ｎｏｖｉｋｏｖ－Ｗｉｔｔｅｎ（缩写为ＷＺＮＷ）模型中加入破坏共形对称性的约束，获得了一个新的经典完全可积的二维场论体系，它把著名的ｓｉｎｈ－Ｇｏｒｄｏｎ方程作为其特例。这个广义ｓｉｎｈ－Ｇｏｒｄｏｎ体系的特点是完全可积性和可超定域化，并且描写这些特点的ｒ矩阵是杨－Ｂａｘｔｅｒ方程（经典的）的一个解，它反对称，依赖于两个谱参数，但通过Ｌｏｏｐ代数的自同构变换和谱参数的重新定义后，此ｒ矩阵仍是依赖于一个谱参数的三角型ｒ矩阵。 关键词：     

The Nondynamical r-Matrix Structure of the Elliptic Calogero–Moser Model

In this Letter, we construct a new Lax operator for the elliptic Calogero–Moser model with N=2. The nondynamical r-matrix structure of this Lax operator is also studied. The relation between our Lax operator and the Lax operator given by Krichever is also obtained.     

三角Calogero-Moser模型的非动力学r矩阵结构

通过一定规范变换,构造了三角Calogero–Moser模型一种新的Lax算子,使其具有相应的非动力学r矩阵结构.同时发现该r矩阵结构与三角Ruijsenaars–Schneider模型的r矩阵完全相同.     

The Nondynamical r-Matrix Structure of the Elliptic Ruijsenaars-Schneider Model with N = 2

We demonstrate that in a certain gauge the elliptic Ruijsenaars-Schneider model with N = 2 admits a nondynamical r-matrix structure and the corresponding classical rmatrix is the same as that of its nonrelativistic counterpart (CalogereMoser model) in the same gauge. The relation between our (classical) Lax operator and the Lax operator given by Ruijsenaars is also obtained.     

The CMS experiment at the LHC-status and physics potential

The most general momentum independent dynamical r-matrices are described for the standard Lax representation of the degenerate Calogero-Moser models based on gl _n and those r-matrices whose dynamical dependence can be gauge d away are selected. In the rational case, a non-dynamical r-matrix resulting from gauge transformation is given explicitly as an antisymmetric solution of the classical Yang-Baxter equation that belongs to the Frobenius subalgebra of gl _n consisting of the matrices with vanishing last row.     

Integrable Rosochatius Deformations of the Restricted cKdV Flows

Three novel finite-dimensional integrable Hamiltonian systems of Rosochatius type and their Lax representations are presented. We make a deformation for the Lax matrbces of the Neumann type, the Bargmann type and the high-order symmetry type of restricted cKdV flows by adding an additional term and then prove that this kind of deformation does not change the r-matrix relations. Finally the new integrable systems are generated from these deformed Lax matrices.     

On the Restricted Toda and c-KdV Flows of Neumann Type

It is proven that on a symplectic submanifold the restricted c-KdV flow is just the interpolating Hamiltonian flow of invariant for the restricted Toda flow, which is an integrable symplectic map of Neumann type. They share the common Lax matrix, dynamical r-matrix and system of involutive conserved integrals. Furthermore, the procedure of separation of variables is considered for the restricted c-KdV flow of Neumann type.     

Strong integrability of the bi-YB–WZ model

Letters in Mathematical Physics - We identify the r-matrix governing the Poisson brackets of the matrix elements of the Lax operator of the bi-YB–WZ model.     

Analytical Bethe Ansatz,Canonical Bäcklund Transformation and Q-Operator For A New Discrete Integrable Heirarchy

A new discrete heiarchy of integrable equations is generated from a new Lax Operator and a canonical Bäcklund transformation of the system is derived using Sklyanin’s formalism, based on the classical r-matrix. By quantising the system a quantum analogue of the corresponding canonical Bäcklund transformation is obtained and certain properties of the associated Q-operator are examined. Finally the analytical Bethe Ansatz is used to solve for the spectrum.     

TOTALLY CLASSICAL CALOGERO MODEL

We show that the standard Calogero Lax matrix can be interpreted as a function on the fuzzy sphere and the Avan–Talon r-matrix as a function on the direct product of two fuzzy spheres. We calculate the limiting Lax function and r-function when the fuzzy sphere tends to the ordinary sphere and we show that they define an integrable model interpreted as a large N Calogero model by Bordemann, Hoppe and Theisen.     

Iterative Construction of U q (s ℓ(n+1)) Representations and Lax Matrix Factorisation

The iterative construction of a generic representation of g?(n + 1) or of the trigonomentric deformation of its enveloping algebra is conveniently formulated in terms of Lax matrices. The Lax matrix of the constructed representation factorises into parts determined by the Lax matrix of a generic representation of the algebra with reduced rank and others appearing in the factorised expression of the Lax matrix of the special Jordan–Schwinger representation.     

Some quasi-periodic solutions to the Kadometsev-Petviashvili and modified Kadometsev-Petviashvili equations

The Kadometsev-Petviashvili (KP) and modified KP (mKP) equations are retrieved from the first two soliton equations of coupled Korteweg-de Vries (cKdV) hierarchy. Based on the nonlinearization of Lax pairs, the KP and mKP equations are ultimately reduced to integrable finite-dimensional Hamiltonian systems in view of the r-matrix theory. Finally, the resulting Hamiltonian flows are linearized in Abel-Jacobi coordinates, such that some specially explicit quasi-periodic solutions to the KP and mKP equations are synchronously given in terms of theta functions through the Jacobi inversion.     

The Classical r-Matrix of AdS/CFT and its Lie Bialgebra Structure

In this paper we investigate the algebraic structure of AdS/CFT in the strong-coupling limit. We propose an expression for the classical r-matrix with (deformed) symmetry, which leads to a quasi-triangular Lie bialgebra as the underlying symmetry algebra. On the fundamental representation our r-matrix coincides with the classical limit of the quantum R-matrix.     

Non-autonomous Hénon-Heiles system from Painlevé class

We show how to deform separable Hénon-Heiles system with isospectral Lax representation, related with the stationary flow of the $5th$-order KdV, to respective non-autonomous systems of Painlevé type with isomonodromic Lax representation.     

The Ablowitz–Ladik hierarchy integrability analysis revisited: the vertex operator solution representation structure

A regular gradient-holonomic approach to studying the Lax type integrability of the Ablowitz–Ladik hierarchy of nonlinear Lax type integrable discrete dynamical systems in the vertex operator representation is presented. The relationship to the Lie-algebraic integrability scheme is analyzed and the connection with the τ-function representation is discussed.     

Lax representation for the systems of S. Kowalevskaya type

We describe a number of dynamical systems that are generalizations of the S. Kowalevskaya system and admit the Lax representation.