Related papers: Multi-Hamiltonian structures for r-matrix systems

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…

Bi-Hamiltonian structures are of great importance in the theory of integrable Hamiltonian systems. The notion of compatibility of symplectic structures is a key aspect of bi-Hamiltonian systems. Because of this, a few different notions of…

We consider three different incompatible bi-Hamiltonian structures for the Lagrange top, which have the same foliation by symplectic leaves. These bivectors may be associated with the different 2-coboundaries in the Poisson-Lichnerowicz…

In this paper we define a family of systems which have similarities with the Toda lattice. We construct two Lax pair representations and the associate Poisson structures for these systems. These systems lie between the classical Toda…

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principle $G$-bundle on a surface. The moduli space is a Poisson variety with Atiyah-Bott Poisson…

We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators. Explicit formulas for Casimir elements of this elliptic Poisson…

This paper provides a geometric description for Lie--Hamilton systems on $\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of…

We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler…

In this work, we conduct a systematic study of Hamiltonian and quasi-Hamiltonian systems within the framework of nondecomposable generalized Poisson geometry. Our focus lies on the interplay between the algebraic structure of…

In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian…

We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson…

In a preceding paper we introduced a notion of compatibility between a Jacobi structure and a Riemannian structure on a smooth manifold. We proved that in the case of fundamental examples of Jacobi structures : Poisson structures, contact…

In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a…

We construct the tri-Hamiltonian structure of the two-dimensional Toda hierarchy using the R-matrix theory.

We show that the space of orthogonally separable coordinates on the sphere $S^3$ induces a natural family of integrable systems, which after symplectic reduction leads to a family of integrable systems on $S^2 \times S^2$. The generic…

For a smooth complex algebraic curve $X$ and a reduced effective divisor $D$ on $X$, we introduce a notion of $D$-level structure on parahoric $\mathcal{G}_{\boldsymbol \theta}$-torsors over $X$, for any connected complex reductive Lie…

We present novel Hamiltonian descriptions of some three-dimensional systems including two well-known systems describing the three-wave-interaction problem and some well-known chaotic systems, namely, the Chen, L\"u, and Qi systems. We show…

We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional {\em quadratic} first integrals, thus constructing a large…

We study the Poisson structure associated to the defocusing Ablowitz-Ladik equation from a functional-analytical point of view, by reexpressing the Poisson bracket in terms of the associated Caratheodory function. Using this expression, we…

Given a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poisson structure on an open neighbourhood of its regular (not necessarily compact)…