Related papers: Full hamiltonian structure for a parametric couple…
This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect…
The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the…
A few 2+1-dimensional equations belonging to the KP and modified KP hierarchies are shown to be sufficient to provide a unified picture of all the integrable cases of the cubic and quartic H\'enon-Heiles Hamiltonians.
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as…
The theory of Poisson Vertex Algebras (PVAs) is a good framework to treat Hamiltonian partial differential equations. A PVA consists of a pair $(\mathcal{A},\{\cdot_\lambda\cdot\})$ of a differential algebra $\mathcal{A}$ and a bilinear…
This note describes a local Poisson structure (up to homotopy) associated to corners in four-dimensional gravity in the coframe (Palatini--Cartan) formalism. This is achieved through the use of the BFV formalism. The corner structure…
Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which…
We find a two-parameter family of ordinary differential systems in dimension five with the affine Weyl group symmetry of type $D_3^{(2)}$. We show its symmetry and holomorphy conditions. This is the second example which gave higher order…
New generalized Poisson structures are introduced by using skew-symmetric contravariant tensors of even order. The corresponding `Jacobi identities' are given by the vanishing of the Schouten-Nijenhuis bracket. As an example, we provide the…
A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum…
The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence…
We introduce a mixed holomorphic-topological gauge theory in three dimensions associated to a (freely generated) Poisson vertex algebra. The $\lambda$-bracket of the PVA plays the role of the structure constants of the gauge algebra and the…
We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$…
We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless…
We establish quadratic Poisson brackets for the generalized Camassa--Holm peakon structure introduced in \cite{AFR23}. The calculation is based on the halving of the spectral parameter dependent $r$-matrix used to define the linear Poisson…
The quasi-integrable KdV equation has been obtained from the corresponding deformation of the Hamiltonian for the usual KdV system. Following suitable gauge-fixing, it has been found that the quasi-conservation condition is satisfied and an…
We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis…
We describe a general procedure which allows to construct, starting from a given Hamiltonian, the whole family of new ones sharing the same set of unparameterized trajectories in phase space. The symmetry structure of this family can be…
The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting $\W$-algebra is a…
An analysis of extension of Hamiltonian operators from lower order to higher order of matrix paves a way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of…