arXiv research

Paper studies invariant distributions of bi-Hamiltonian structures.

Hydrodynamic structures linked to F-manifolds.

We show that the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym hierarchies can be obtained by applying a reduction process to a simple Poisson pair defined on the loop algebra of $\mathfrak{sl}(2,\mathbb{R})$. The reduction process is a bi-Hamiltonian reduction, that can be canonically performed on every …

We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there ex…

A bi-Hamiltonian structure is a pair of Poisson structures $\mathcal P$, $\mathcal Q$ which are compatible, meaning that any linear combination $α\mathcal P + β\mathcal Q$ is again a Poisson structure. A bi-Hamiltonian structure $(\mathcal P, \mathcal Q)$ is called flat if $\mathcal P$ and $\mathcal Q$ can be simultane…

Arnold-Liouville systems cannot be bi-Hamiltonian generically.

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being …

The paper connects a second order ODE to Sasakian structures and bi-Hamiltonian systems.

Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_τ(P)$ along a vector field $τ$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_τ^2(P),P]=0$, where $[\cdot,\cdot]$ is the Schouten bracket. We further prove that…

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 compatibility have been introduced. In this paper we show that, under some additi…

Recently S.A. Merkulov established a link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of minimal resolutions of props. In particular he described the prop profile of Poisson geometry. In this paper we define a prop such that represen…

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropr…

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theo…

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagran…

Integrable geodesics found on special orthogonal group.

Paper proves vanishing terms in a second Poisson bracket for a specific system.

Paper bridges quantum and classical mechanics for open systems.

Hamiltonian formulation of N=3 systems is considered in general. The most general solution of the Jacobi equation in ${\mathbb R}^3$ is proposed. Compatible Poisson structures and the corresponding bi-Hamiltonian N=3 systems are also discussed.

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarch…

Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.

Supergeneralization of $\DC P(N)$ provided by even and odd Kählerian structures from Hamiltonian reduction are construct.Operator $Δ$ which used in Batalin-- Vilkovisky quantization formalism and mechanics which are bi-Hamiltonian under corresponding even and odd Poisson brackets are considered.

The paper explores geometric aspects of Miura transformations in integrable systems.

Proves local bi-integrability of bi-Hamiltonian systems on real smooth manifolds.

We discuss the polynomial bi-Hamiltonian structures for the Kowalevski top in special case of zero square integral. An explicit procedure to find variables of separation and separation relations is considered in detail.

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 Kupershmidt's deformation of a bi-Hamiltonian system.

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 …

We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of Darboux coordinates defined by the eigenvalues and the eigenvectors of the Lax matri…

In the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized Stäckel systems and, as a particular case, of the quasi-bi-Hamiltonian systems. As an application, we recover the Haantjes manifolds for the rational Calogero model with three particles and for the Benenti…

We use refined spectral sequence arguments to calculate known and previously unknown bi-Hamiltonian cohomology groups, which govern the deformation theory of semi-simple bi-Hamiltonian pencils of hydrodynamic type with one independent and \( N\) dependent variables. In particular, we rederive the result of Dubrovin-Liu…

We prove that the Kupershmidt deformation of a bi-Hamiltonian system is itself bi-Hamiltonian. Moreover, Magri hierarchies of the initial system give rise to Magri hierarchies of Kupershmidt deformations as well. Since Kupershmidt deformations are not written in evolution form, we start with an outline a geometric fram…

We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion w…

We give a characterisation of Atiyah's and Hitchin's transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole modu…

We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p_d(\hat{F}, d_1,d_2)$ is isomorphic to $\mathbb{R}$ for $(p,d)=(0,0)$, to $C^\infty (\mathbb{R})$ for $(p,d)=(1,1)$, $(2,1)$, $(…

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.

We give the following results for Pinkall's central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explic…

The purpose of this note is to give a simple description of a (complete) family of functions in involution on certain hermitian symmetric spaces. This family, obtained via bi-hamiltonian approach using the Bruhat Poisson structure, is especially simple for the projective spaces, where the formulas in terms of the momen…

We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,π)$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is equipped with a bi-Hamil…

We briefly recall the history of the Nijenhuis torsion of (1,1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic-Haantjes structures of Tempesta and Tondo generalize the classical approach to integrable systems in the bi-hamiltonian and s…

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 of a natural symplectic structure on the cotangent bundle. First it is developed …

Constructs tri-Hamiltonian structure and Frobenius manifold for asymmetric gAL hierarchy

Local tri-Hamiltonian structure for Ablowitz-Ladik hierarchy established.

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescali…

The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair $A_1$, $A_2$, where $A_1$ is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equiv…

Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…

We observe that the modular class of a Poisson-Nijhenhuis manifold has a canonical representative and that, under a cohomological assumption, this vector field is bi-hamiltonian. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies.

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the $n$-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on …

We show that discrete $W_m$ lattices are bi-Hamiltonian, using geometric realizations of discretizations of the Adler-Gel'fand-Dikii flows as local evolutions of arc length-parametrized polygons in centro-affine space. We prove the compatibility of two known Hamiltonian structure defined on the space of geometric invar…

We present a simple way of generating the infinite set of Jacobi tensors, compatible with a given one, via the "gauge transformations" of the functions on Jacobi manifold. We consider also some applications of this result to the construction of bi-Hamiltonian systems on Jacobi manifolds.