Related papers: Birkhoff normal form via decorated trees

We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached $L_\infty$-algebra controlling its…

We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type…

We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems. The…

Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,...) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is…

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the…

We study a family of tree-type diagrams that arise in studies of the cumulant expansion in discrete Erd\H os-R\'enyi random matrix models. Using a version of the Pr\" ufer code, we obtain an explicit expression for the number of tree-type…

We introduce discrete equational theories where operations are induced by those having discrete arities. We characterize the corresponding monads as monads preserving surjections. Using it, we prove Birkhoff type theorems for categories of…

It is shown how derived brackets naturally arise in sigma-models via Poisson- or antibracket, generalizing a recent observation by Alekseev and Strobl. On the way to a precise formulation of this relation, an explicit coordinate expression…

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term…

We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector…

Let k be a local field and let A be the two-by-two matrix algebra over k. In our previous work we developed a theory that allows the computation of the set of maximal orders in A containing a given suborder. This set is given as a sub-tree…

This work is devoted to the establishment of a Poisson structure for a format of equations known as Generalized Lotka-Volterra systems. These equations, which include the classical Lotka-Volterra systems as a particular case, have been…

We introduce the concept of braided noncommutative Poisson bialgebras. The theory of cocycle bicrossproducts for noncommutative Poisson bialgebras is developed. As an application, we solve the extending problem by using some non-abelian…

A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.

We describe a method for calculating action-angle variables in axisymmetric galactic potentials using Birkhoff normalization, a technique from Hamiltonian perturbation theory. An advantageous feature of this method is that it yields…

We give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique…

The construction of HNN-extensions of involutive Hom-associative algebras and involutive Hom-Lie algebras is described. Then, as an application of HNN-extension, by using the validity of Poincar\'e-Birkhoff-Witt theorem for involutive…

Stable Hamiltonian structures generalize contact forms and define a volume-preserving vector field known as the Reeb vector field. We study two aspects of Reeb vector fields defined by stable Hamiltonian structures on 3-manifolds: on one…

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the Lie algebra of U, \g=\u^\C. Each…

We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa-Holm equation, and…