arXiv subjects

Andriy Panasyuk

Publications and source records attributed to Andriy Panasyuk.

15 recordsLinked to original sources

arXiv2026

Dispersionless Hirota system and hidden symmetries of heavenly equation

In 2021 Konopelchenko, Schief and Szereszewski observed that solutions of 4D dispersionless Hirota system also solve the general heavenly equation describing self-dual vacuum Einstein metrics in neutral signature. They also noticed that the symmetry $f\mapsto \Phi(f)$ of the Hirota system essentially changes the properties of the corresponding metric. In this paper we restate these observations in the context of I and II Pleba\'nski heavenly equation (I,II PHE). Namely, we first find 5D analogues of these equations. We then consider a special type of symmetry generalizing the so-called tri-holomorphic symmetry of I or II PHE. The reduction with respect to this symmetry (which in a sense imitates the reduction of self-dual vacuum Einstein metrics with respect to a tri-holomorphic symmetry ending in special Einstein--Weyl structures) gives an analogue of the dispersionless Hirota system for I and II PHE. Such a point of view allows to reinterpret the symmetry $f\mapsto \Phi(f)$ mentioned and obtain explicit formulas for the metric depending on $\Phi$. We present some examples showing how the Weyl spinor changes along with $\Phi$.

math-ph↗

arXiv2025

On compatible linear and quadratic Poisson brackets on $gl(N)$

In the present paper, using two constant tensors $c$ and $b$ on $sl(N)\otimes sl(N)$ satisfying certain linear-quadratic equation and a technique of Poisson bivectors and Schouten brackets, we explicitly construct quadratic Poisson bracket on the space $ (sl(N)+\mathbb{C})^*$ which is compatible with the standard Lie--Poisson bracket on $gl(N)^* \simeq (sl(N)\oplus \mathbb{C})^*$. The case of $N=3$ is considered in details. The relation of the proposed brackets with the generalized classical Sklyanin algebras is explained.

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arXiv2025

Rational interpolants and solutions of dispersionless Hirota system

The aim of this paper is to construct a class of explicit nontrivial rational solutions of the dispersionless Hirota system of PDEs. All the solutions in this class are of homogeneity degree 1 and are quotients of homogeneous polynomials. It is well-known that the solutions of the Hirota dispersionless systems describe Veronese webs. By nontriviality of the solutions it is meant that the corresponding Veronese webs are nonflat at generic points.

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arXiv2022

Webs, Nijenhuis operators, and heavenly PDEs

In 1989 Mason and Newman proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields $X_1(\lambda),X_2(\lambda)$ which are divergence free with respect to some volume form. Earlier (in 1975) Pleba\'nski showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Pleba\'nski heavenly PDEs. Other PDEs leading to Mason--Newman vector fields are also known in the literature: Husain--Park (1992--94), Schief (1996). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new ``heavenly'' PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko--Schief--Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the ``heavelny'' PDEs and the existence of related vacuum Einstein metrics in 4D-case.

math.DG↗

arXiv2019

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that there exists a 10-parametric family of quadratic Poisson structures on $\gl(3)^*$ compatible with the canonical linear Poisson structure and containing the 3-parametric family of quadratic bivectors recently introduced by Vladimir Sokolov. The involutive family of polynomial functions related to the corresponding Poisson pencils contains the hamiltonian of the polynomial form of the elliptic Calogero--Moser system.

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arXiv2019

Invariant Nijenhuis Tensors and Integrable Geodesic Flows

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $Π_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $Π_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$.

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arXiv2017

Kronecker webs, Nijenhuis operators, and nonlinear PDEs

The aim of this paper is two-fold. First, a survey of the theory of Kronecker webs and their relations with bihamiltonian structures and PDEs is presented. Second, a partial solution to the problem of bisymplectic realization of a bihamiltonian structure is given. Both the goals are achieved by means of the notion of a partial Nijenhuis operator, which is studied in detail.

math.DG↗

arXiv2016

Dirac brackets and reduction of invariant bi-Poisson structures

Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $ω_1|_{X^r_\mathfrak{h}}$ and $ω_2|_{X^r_\mathfrak{h}}$ generates an $N(H^0)/H^0$-invariant bi-Poisson structure on $X^r_\mathfrak{h}$, where $N(H^0)$ is the normalizer in $G$ of the identity component $H^0$ of $H$. The action of $\widetilde G=N(H^0)/H^0$ on $X^r_\mathfrak{h}$ is locally free and proper and, moreover, the spaces $A^G$ of $G$-invariant functions on $X$ and $A^{\widetilde G}$ of $\widetilde G$-invariant functions on $X^r_\mathfrak{h}$ can be canonically identified and therefore the bi-Poisson structure $\{(η^t)'\}$ induced on $A^G\simeq A^{\widetilde G}$ can be treated as the reduction with respect to a {\em locally free} action of a Lie group which essentially simplifies the study of $\{(η^t)'\}$.

math.DG↗

arXiv2016

Veronese webs and nonlinear PDEs

Veronese webs are closely related to bi-Hamiltonian systems, as was shown by Gelfand and Zakharevich. Recently a correspondence between Veronese three-dimensional webs and three-dimensional Einstein-Weyl structures of hyper-CR type was established. The latter were parametrized by Dunajski and Krynski via the solutions of the dispersionless Hirota equation. In this paper we show relations of Veronese three-dimensional webs to several other integrable equations, deform these equations preserving integrability via a dispersionless Lax pair and compute the corresponding contact symmetries, Backlund transformations and Einstein-Weyl structures. Realization of Veronese webs through solutions of these deformed integrable PDE is based on a correspondence between partially integrable Nijenhuis operators to the operator fields with vanishing Nijenhuis tensor. This correspondence could be used to construct a link between bi-Hamiltonian finite-dimensional integrable systems and dispersionless integrable PDE related to the Veronese webs.

math.DG↗

arXiv2012

Lie--Poisson pencils related to semisimple Lie algebras: towards classification

Let $\mathfrak{g}$ be a vector space and $[,],[,]'$ be a pair of Lie brackets on $\mathfrak{g}$. By definition they are compatible if $[,]+[,]'$ is again a Lie bracket. Such pairs play important role in bihamiltonian and $r$-matrix formalisms in the theory of integrable systems. We propose an approach to a long standing problem of classification of such pairs in the case when one of them, say $[,]$, is semisimple. It is known that any such pair is determined by a linear operator on $(\mathfrak{g},[,])$, which is defined up to adding a derivation. We propose a special fixing of this operator to get rid of this ambiguity and consider the operators preserving the root decomposition with respect to a Cartan subalgebra. The classification leads to two disjoint classes of pairs depending on the symmetry properties of the corresponding operator with respect to the Killing form. Within each class we recover known examples and obtain new ones. We present a list of examples in each case and conjecture the completeness of these lists.

math.DG↗

arXiv2005

Algebraic Nijenhuis operators and Kronecker Poisson pencils

We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on $\frak{g}^*$ related to an algebraic Nijenhuis operator $N:\frak{g}\to \frak{g}$ on a finite-dimensional Lie algebra $\frak{g}$. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related to Borel subalgebras and a new proof of the complete integrability of the free rigid body system on $\frak{gl}_n$.

math.DG↗

arXiv2002

On integrability of generalized Veronese curves of distributions

Given a 1-parameter family of 1-forms $\g(t)= \g_0+t\g_1+...+t^n\g_n$, consider the condition $d\g(t)\wedge\g(t)=0$ (of integrability for the annihilated by $\g(t)$ distribution $w(t)$). We prove that in order that this condition is satisfied for any $t$ it is sufficient that it is satisfied for $N=n+3$ different values of $t$ (the corresponding implication for $N=2n+1$ is obvious). In fact we give a stronger result dealing with distributions of higher codimension. This result is related to the so-called Veronese webs and can be applied in the theory of bihamiltonian structures.

math.DG↗

arXiv2002

Projections of Jordan bi-Poisson structures that are Kronecker, diagonal actions, and the classical Gaudin systems

We propose a method of constructing completely integrable systems based on reduction of bihamiltonian structures. More precisely, we give an easily checkable necessary and sufficient conditions for the micro-kroneckerity of the reduction (performed with respect to a special type action of a Lie group) of micro-Jordan bihamiltonian structures whose Nijenhuis tensor has constant eigenvalues. The method is applied to the diagonal action of a Lie group $G$ on a direct product of $N$ coadjoint orbits $Ø=O_1\times...\times O_N$ endowed with a bihamiltonian structure whose first generator is the standard symplectic form on $Ø$. As a result we get the so called classical Gaudin system on $Ø$. The method works for a wide class of Lie algebras including the semisimple ones and for a large class of orbits including the generic ones and the semisimple ones.

math.DG↗

arXiv2000

Symplectic realizations of bihamiltonian structures

A method of constructing a class of bihamiltonian structures is presented. Elements of this class are generalizations of the so-called bihamiltonian structures of general position on odd-dimensional manifolds. The method consists in a simultaneous reduction of both the real and imaginary parts of a complex symplectic form. Necessary and sufficient conditions of getting a bihamiltonian structure from the mentioned class are obtained. The second part of the paper is devoted to a series of examples of such a reduction related to semisimple Lie algebras.

math.DG↗

arXiv2000

Veronese webs for bihamiltonian structures of higher corank

It is shown how the well-known class of bihamiltonian structures in general position can be extended to a wider class. A generalization of the corresponding notion of a Veronese web for this wider class is presented (in the general position case Veronese webs form complete systems of local invariants for bihamiltonian structures). Some examples are considered.

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