Related papers: Dynamical R-Matrices for Integrable Maps
The Calogero type matrix discretization scheme is applied to constructing the Lax type integrable discretizations of one wide enough class of nonlinear integrable dynamical systems on functional manifolds. Their Lie-algebraic structure and…
The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). We…
The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter…
The motion of a rigid body in a quadratic potential is an important example of an integrable Hamiltonian system on a dual to a semidirect product Lie algebra so(n) x Symm(n). We give a Lagrangian derivation of the corresponding equations of…
Density-matrix topology, defined through the geometric property of the relevant modular Hamiltonian, can undergo transitions in the corresponding open-system dynamics. While symmetry considerations are crucial to ensure such a dynamic…
Determining whether a dynamical system is integrable is generally a difficult task which is currently done on a case by case basis requiring large human input. Here we propose and test an automated method to search for the existence of…
Working in a symplectic reduction framework, we construct a dynamical r-matrix for the classical hyperbolic BC(n) Sutherland model with three independent coupling constants. We also examine the Lax representation of the dynamics and its…
Rational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multi-Hamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical R-matrix approach…
We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that…
We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow…
A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with…
The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear…
The integrable system is introduced based on the Poisson $ rs $-matrix structure. This is a generalization of the Gaudin magnet, and in SL(2) case isomorphic to the generalized Neumann model. The separation of variables is discussed for…
A new Lax representation for the Bogoyavlensky lattice is found, its $r$--matrix interpretation is elaborated. The $r$--matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of associative…
We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding transfer matrices give rise to time evolution equations for the initial Lax…
In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew's triple and…
The local symplectic theory of integrable systems is fundamental to understand their global theory, as well as the behavior near singularities of fundamental models from classical and quantum mechanics which are known to be integrable, such…
We propose realizations of the Poisson structures for the Lax representations of three integrable $n$-body peakon equations, Camassa--Holm, Degasperis--Procesi and Novikov. The Poisson structures derived from the integrability structures of…
The first part of the present paper is devoted to a systematic construction of continuous-time finite-dimensional integrable systems arising from the rational su(2) Gaudin model through certain contraction procedures. In the second part, we…
Within the class of integrable Calogero models associated with (semi-)simple Lie algebras and with symmetric pairs of Lie algebras identified in a previous paper, we analyze whether and to what extent it is possible to find a gauge…