Poisson Brackets Of Hydrodynamic Type Research Articles

Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinite-dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension ⩽n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

Poisson Brackets of Hydrodynamic Type and Their Generalizations

In this paper, we consider Hamiltonian structures of hydrodynamic type and some of their generalizations. In particular, we discuss the questions concerning the structure and special forms of the corresponding Poisson brackets and the connection of such structures with the theory of integration of systems of hydrodynamic type.

English

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we show that fermionic Novikov algebras equipped with invariant non-degenerate symmetric bilinear forms are Novikov algebras.

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

The theory of multidimensional Poisson vertex algebras provides a completely algebraic formalism for studying the Hamiltonian structure of partial differential equations for any number of dependent and independent variables. We compute the cohomology of the Poisson vertex algebras associated with twodimensional, two-component Poisson brackets of hydrodynamic type at the third differential degree. This allows obtaining their corresponding Poisson–Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of nonequivalent infinitesimal deformations, nor in the third group, corresponding to the obstructions to extending such deformations.

On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

The theory of Poisson vertex algebras (PVAs) (Barakat et al. in Jpn J Math 4(2):141–252, 2009) 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 operation called the $${\lambda}$$ -bracket. We extend the definition to the class of algebras $${\mathcal{A}}$$ endowed with $${d \geq 1}$$ commuting derivations. We call this structure a multidimensional PVA: it is a suitable setting to study Hamiltonian PDEs with d spatial dimensions. We apply this theory to the study of symmetries and deformations of the Poisson brackets of hydrodynamic type for d = 2.

A Compendium of Lie Structures on Tensor Products

We demonstrate how a simple linear-algebraic technique used earlier to compute low-degree cohomology of current Lie algebras, can be utilized to compute other kinds of structures on such Lie algebras, and discuss further generalizations, applications, and related questions. While doing so, we touch upon such seemingly diverse topics as associative algebras of infinite representation type, Hom-Lie structures, Poisson brackets of hydrodynamic type, Novikov algebras, simple Lie algebras in small characteristics, and Koszul dual operads.

Underlying lie algebras of quadratic Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and the Hamiltonian operators in formal variational calculus. In this note we prove that the underlying Lie algebras of quadratic Novikov algebras are 2-step nilpotent. Moreover, we give the classification up to dimension 10.

A twisted generalization of linear Poisson brackets of hydrodynamic type

We study a twisted generalization of linear Poisson brackets of hydrodynamic type, in which the Lie bracket is replaced by a Hom–Lie bracket. With some natural additional conditions, such structures correspond to the Hom–Novikov algebras introduced by Yau, which is the twisted version of Balinsky–Novikov's approach of constructing a Lie algebra from a Novikov algebra. Certain central extensions of this twisted generalization of linear Poisson brackets of hydrodynamic type are obtained from the bilinear forms on their corresponding Hom–Novikov algebras satisfying some invariance conditions. Finally, we give some examples of the infinite-dimensional Hom–Lie algebras constructed from the Hom–Novikov algebras. In particular, there is an interesting twisted generalization of the Virasoro algebra.

Hamiltonian Systems of Hydrodynamic Type in 2 + 1 Dimensions

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 Lax pairs and an infinity of hydrodynamic reductions.

Form-invariant Poisson brackets of hydrodynamic type with several spatial variables

Form-invariant Poisson brackets of hydrodynamic type with several spatial variables x1,…,xp are introduced for any commuting vector fields U1,…,Um on a manifold Mn. The incompressibility equation for certain parameters of the Poisson brackets is derived. Poisson brackets connected with an arbitrary dynamical system on Mn are constructed. A gauge transform for the Poisson brackets with one spatial variable x is defined and its action onto the tensor invariants is studied. A general identity between the coefficients bkij(u) and gij(u) is found.

The classification of nonsingular multidimensional Dubrovin-Novikov brackets

In this paper, the well-known Dubrovin-Novikov problem posed as long ago as in 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. In this paper, we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differential-geometric methods. A key role in the solution of this problem is played by the theory of compatible metrics earlier constructed by the present author.

Tensor Invariants of the Poisson Brackets of Hydrodynamic Type

Form-invariant solutions for the Poisson brackets of hydrodynamic type on a manifold Mn with (2,0)-tensor gij(u) of rank m ≤ n are derived. Tensor invariants of the Poisson brackets are introduced that include a vector field V (or dynamical system V) on Mn, the Lie derivative LV gij and symmetric (k, 0)-tensors \(h^{ij\cdots\ell}\). Several scalar invariants of the Poisson brackets are defined. A nilpotent Lie algebra structure is disclosed in the space of 1-forms \({\mathcal{A}}_u \subset T^*_u(M^n)\) that annihilate the (2,0)-tensor gij(u). Applications to the one-dimensional gas dynamics are presented.

Bilinear forms on fermionic Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension ⩽4. Motivated by the one in dimension 4, we construct some examples in high dimensions.

Invariant dynamical system for a Poisson bracket of hydrodynamic type

An invariant dynamical system V is introduced for any Poisson bracket of hydrodynamic type on M n with ( 2 , 0 ) -tensor g i j ( u ) of rank m < n . The dynamical system V is applied for constructing non-constant invariants of the Poisson bracket. Invariant ( n − m ) -dimensional nilpotent Lie algebras A u are found that are embedded into the cotangent spaces T u ∗ ( M n ) .

Invariant foliations for the Poisson brackets of hydrodynamic type

Abstract An invariant foliation F with an induced non-degenerate metric 〈 v , w 〉 of constant curvature K is discovered for any degenerate Poisson bracket of hydrodynamic type. A hidden structure of nilpotent Lie algebras in the cotangent spaces is disclosed for g i j ( u ) = 0 .

On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type

A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parameterize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the one hand, two realizations of the first negative flow of the AKNS model and, on the other, two-component generalizations of Camassa-Holm and Dym type equations. The two-component generalizations of Camassa-Holm and Dym type equations can be obtained from the negative order Hamiltonians constructed from the Lenard relations recursively applied on the Casimir of the first Poisson bracket of hydrodynamic type. The positive order Hamiltonians, which follow from Lenard scheme applied on the Casimir of the second Poisson bracket of hydrodynamic type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS Hamiltonians give rise to charges conserved with respect to equations of motion of two-component Camassa-Holm and two-component Dym type equations.

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.

Some Results on Novikov–Poisson Algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. A Novikov–Poisson algebra is a Novikov algebra with a compatible commutative associative algebraic structure, which was introduced to construct the tensor product of two Novikov algebras. In this paper, we commence a study of finite-dimensional Novikov–Poisson algebras. We show the commutative associative operation in a Novikov–Poisson algebra is a compatible global deformation of the associated Novikov algebra. We also discuss how to classify Novikov–Poisson algebras. And as an example, we give the classification of 2-dimensional Novikov–Poisson algebras.

The automorphisms of Novikov algebras in low dimensions

Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. They also correspond to a class of vertex algebras. An automorphism of a Novikov algebra is a linear isomorphism φ satisfying φ(xy) = φ(x)φ(y) which keeps the algebraic structure. The set of automorphisms of a Novikov algebra is a Lie group whose Lie algebra is just the Novikov algebra's derivation algebra. The theory of automorphisms plays an important role in the study of Novikov algebras. In this paper, we study the automorphisms of Novikov algebras. We get some results on their properties and classification in low dimensions. These results are fundamental in a certain sense, and they will serve as a guide for further development. Moreover, we apply these results to classify Gel'fand–Dorfman bialgebras and Novikov–Poisson algebras. These results also can be used to study certain phase spaces and geometric classical r-matrices.

The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.