Related papers: Dynamical R-Matrices for Integrable Maps
A class of Poisson embeddings of reduced, finite dimensional symplectic vector spaces into the dual space $\Lg_R^*$ of a loop algebra, with Lie Poisson structure determined by the classical split $R$--matrix $R=P_+ - P_-$ is introduced.…
The integrability of a family of hamiltonian systems, describing in a particular case the motionof N ``peakons" (special solutions of the so-called Camassa-Holm equation) is established in the framework of the $r$-matrix approach, starting…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…
We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the…
We use the r-matrix formulation to show the integrability of geodesic flow on an $N$-dimensional space with coordinates $q_k$, with $k=1,...,N$, equipped with the co-metric $g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)$. This flow is…
The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The…
In this paper we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for…
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…
We consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical $r$-matrix, first introduced by Sklyanin as the classical analogue of the quantum…
We establish the algebraic origin of the following observations made previously by the authors and coworkers: (i) A given integrable PDE in $1+1$ dimensions within the Zakharov-Shabat scheme related to a Lax pair can be cast in two…
We show here the separability of Hamilton-Jacobi equation for a hierarchy of integrable Hamiltonian systems obtained from the constrained flows of the Jaulent-Miodek hierarchy. The classical Poisson structure for these Hamiltonian systems…
We develop the approach to the problem of integrable discretization based on the notion of $r$--matrix hierarchies. One of its basic features is the coincidence of Lax matrices of discretized systems with the Lax matrices of the underlying…
We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models…
We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties $V_{n,r}$. The systems are integrable in the non-commutative sense, and by applying a $2r\times…
The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a…
An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The…
For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral…
We study vector quadrirational Yang-Baxter maps representing the momentum-energy transformation of two particles after elastic relativistic collisions. The collision maps admit Lax representations compatible with an r-matrix Poisson…
We describe all linear operators which maps $n-1$-dimensional simplex of idempotent measures to itself. Such operators divided to two classes: the first class contains all $n\times n$-matrices with non-negative entries which has at least…
The canonical Poisson structure of nonlinear sigma-model is presented as a Lie-Poisson r-matrix bracket on coadjoint orbits. It is shown that the Poisson structure of this model is determined by some `hidden singularities' of the Lax…