bi-Hamiltonian Systems Research Articles

On the integrability of three two-component bi-Hamiltonian systems

On the integrability of three two-component bi-Hamiltonian systems

Integrable Bi-Hamiltonian Systems by Jacobi Structure on Real Three-Dimensional Lie Groups

By Poissonization of Jacobi structures on real three-dimensional Lie groups {textbf{G}} and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on {textbf{G}} otimes {mathbb {R}}.

Recursion operators and bi-Hamiltonian representations of cubic evolutionary (2+1)-dimensional systems

We construct all (2+1)-dimensional PDEs depending only on 2nd-order derivatives of unknown which have the Euler–Lagrange form and determine the corresponding Lagrangians. We convert these equations and their Lagrangians to two-component forms and find Hamiltonian representations of all these systems using Dirac’s theory of constraints. We consider three-parameter integrable equations that are cubic in partial derivatives of the unknown applying our method of skew factorization of the symmetry condition. Lax pairs and recursion relations for symmetries are determined both for one-component and two-component forms. For cubic three-parameter equations in the two-component form we obtain recursion operators in 2 × 2 matrix form and bi-Hamiltonian representations, thus discovering three new bi-Hamiltonian (2+1) systems.

Jordan-Kronecker invariants of Borel subalgebras of semisimple Lie algebras

In the theory of bi-Hamiltonian systems, the generalized Mischenko–Fomenko conjecture is known. The conjecture states that there exists a complete set of polynomial functions in involution with respect to a pair of naturally defined Poisson structures on a dual space of a Lie algebra. This conjecture is closely related to the argument shift method proposed by A. S. Mishchenko and A. T. Fomenko in [10]. In research works devoted to this conjecture, a connection was found between the existence of a complete set in bi-involution and the algebraic type of the pencil of compatible Poisson brackets, defined by a linear and constant bracket. The numbers that describe the algebraic type of the generic pencil of brackets on the dual space to a Lie algebra are called the Jordan–Kronecker invariants of a Lie algebra. The notion of Jordan–Kronecker invariants was introduced by A. V. Bolsinov and P. Zhang in [2]. For some classes of Lie algebras (for example, semisimple Lie algebras and Lie algebras of low dimension), the Jordan–Kronecker invariants have been computed, but in the general case the problem of computation of the Jordan–Kronecker invariants for an arbitrary Lie algebra remains open. The problem of computation of the Jordan–Kronecker invariants is frequently mentioned among the most interesting unsolved problems in the theory of integrable systems [4, 5, 6, 11]. In this paper, we compute the Jordan–Kronecker invariants for the series 𝐵𝑠𝑝(2𝑛) and construct complete sets of polynomials in bi-involution for each algebra of the series. Also, we calculate the Jordan–Kronecker invariants for the Borel subalgebras 𝐵𝑠𝑜(𝑛) for any 𝑛. Thus, together with the results obtained in [2] for 𝐵𝑠𝑙(𝑛), this paper presents a solution to the problem of computation of Jordan–Kronecker invariants for Borel subalgebras of classical Lie algebras.

Compatible Poisson Structures and bi-Hamiltonian Systems Related to Low-dimensional Lie Algebras

In this work, we give a method to construct compatible Poisson structures on Lie groups by means of structure constants of their Lie algebras and some vector field. In this way we calculate some compatible Poisson structures on low-dimensional Lie groups. Then, using a theorem by Magri and Morosi, we obtain new integrable bi-Hamiltonian systems with two-, four- and six-dimensional symplectic real Lie groups as phase spaces.

Some bi-Hamiltonian Systems and their Separation of Variables on 4-dimensional Real Lie Groups

In this work, we discuss bi-Hamiltonian structures on a family of integrable systems on 4-dimensional real Lie groups. By constructing the corresponding control matrix for this family of bi-Hamiltonian structures, we obtain an explicit process for finding the variables of separation and the separated relations in detail.

Bi-Hamiltonian Systems in (2+1) and Higher Dimensions Defined by Novikov Algebras

The results from the article [Strachan I.A.B., Szablikowski B.M., Stud. Appl. Math. 133 (2014), 84-117] are extended over consideration of central extensions allowing the introducing of additional independent variables. Algebraic conditions associated to the first-order central extension with respect to additional independent variables are derived. As result $(2+1)$- and, in principle, higher-dimensional multicomponent bi-Hamiltonian systems are constructed. Necessary classification of the central extensions for low-dimensional Novikov algebras is performed and the theory is illustrated by significant $(2+1)$- and $(3+1)$-dimensional examples.

Symmetries, integrals and hierarchies of new (3+1)-dimensional bi-Hamiltonian systems of Monge–Ampère type

We study point symmetries, the corresponding conserved densities and hierarchies of four new bi-Hamiltonian heavenly systems in 3+1 dimensions which we discovered recently. We exhibit an important role played by the inverse recursion operators in the description of the hierarchies. Their use is twofold, either to provide the correct bi-Hamiltonian representation or to generate nonlocal symmetry flows. Invariant solutions w.r.t. nonlocal symmetries will generate (anti-)self-dual gravitational metrics which do not admit Killing vectors which is a characteristic feature of K3 gravitational instanton.

Lax pairs, recursion operators and bi-Hamiltonian representations of (3+1)-dimensional Hirota type equations

We consider (3+1)-dimensional second-order evolutionary PDEs where the unknown u enters only in the form of the 2nd-order partial derivatives. For such equations which possess a Lagrangian, we show that all of them have the symplectic Monge–Ampère form and determine their Lagrangians. We develop a calculus for transforming the symmetry condition to the “skew-factorized” form from which we immediately extract Lax pairs and recursion relations for symmetries, thus showing that all such equations are integrable in the traditional sense. We convert these equations together with their Lagrangians to a two-component form and obtain recursion operators in a 2 × 2 matrix form. We transform our equations from Lagrangian to Hamiltonian form by using the Dirac’s theory of constraints. Composing recursion operators with the Hamiltonian operators we obtain the second Hamiltonian form of our systems, thus showing that they are bi-Hamiltonian systems integrable in the sense of Magri. By this approach, we obtain five new bi-Hamiltonian multi-parameter systems in (3+1) dimensions.

Some super systems with local bi-Hamiltonian operators

For a class of super skew-symmetric operators with eight parameters, two kinds of super Hamiltonian operators are identified by the method of functional multi-vectors. Appropriate decompositions of these operators lead to compatible super Hamiltonian pairs which in turn produce nonlinear super bi-Hamiltonian systems from the trivial x-translation flow. As examples, besides the super Korteweg–de Vries equations and other known ones, new super generalizations are obtained for the Riemann equation, the Hunter–Saxton equation and the Camassa–Holm equation, all of them admit two compatible local super Hamiltonian operators.

Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras

Abstract We prove new theorems related to the construction of the shallow water bi-Hamiltonian systems associated to the semi-direct product of Virasoro and affine Kac–Moody Lie algebras. We discuss associated Verma modules, coadjoint orbits, Casimir functions, and bi-Hamiltonian systems.

Flat F-manifolds, Miura invariants, and integrable systems of conservation laws

We extend some of the results proved for scalar equations in [3,4], to the case of systems of integrable conservation laws. In particular, for such systems we prove that the eigenvalues of a matrix obtained from the quasilinear part of the system are invariants under Miura transformations and we show how these invariants are related to dispersion relations. Furthermore, focusing on one-parameter families of dispersionless systems of integrable conservation laws associated to the Coxeter groups of rank $2$ found in [1], we study the corresponding integrable deformations up to order $2$ in the deformation parameter $\epsilon$. Each family contains both bi-Hamiltonian and non-Hamiltonian systems of conservation laws and therefore we use it to probe to which extent the properties of the dispersionless limit impact the nature and the existence of integrable deformations. It turns out that a part two values of the parameter all deformations of order one in $\epsilon$ are Miura-trivial, while all those of order two in $\epsilon$ are essentially parameterized by two arbitrary functions of single variables (the Riemann invariants) both in the bi-Hamiltonian and in the non-Hamiltonian case. In the two remaining cases, due to the existence of non-trivial first order deformations, there is an additional functional parameter.

Bi-Hamiltonian structures of KdV type**To Franco Magri on the occasion of his 70th birthday, with friendship and admiration.

Combining an old idea of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian operators whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi-parametric families of bi-Hamiltonian systems. We recover known examples and we find apparently new integrable systems whose central invariants are non-zero; this shows that new examples are not Miura-trivial.

New bi-Hamiltonian systems on the plane

We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and recursion operators are also presented in the framework of the Jacobi method.

Poisson–Lie groups, bi-Hamiltonian systems and integrable deformations

Given a Lie–Poisson completely integrable bi-Hamiltonian system on , we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie–Poisson system on a non-abelian Poisson–Lie group of dimension n, where is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson–Lie) Hamiltonian structures on that underly the dynamics of the deformed system and by making use of the group law on , one may obtain two completely integrable Hamiltonian systems on . By construction, both systems admit reduction, via the multiplication in , to the deformed bi-Hamiltonian system in . The previous approach is applied to two relevant Lie–Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems.

Some compatible Poisson structures and integrable bi-Hamiltonian systems on four dimensional and nilpotent six dimensional symplectic real Lie groups

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 theorem we obtain new bi-Hamiltonian systems with four dimensional and nilpotent six dimensional symplectic real Lie groups as phase spaces.

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.

On the Relationship between Two Notions of Compatibility for Bi-Hamiltonian Systems

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 additional assumptions, compatibility in the sense of Magri implies a notion of compatibility due to Fass\`o and Ratiu, that we dub bi-affine compatibility. We present two proofs of this fact. The first one uses the uniqueness of the connection parallelizing all the Hamiltonian vector fields tangent to the leaves of a Lagrangian foliation. The second proof uses Darboux-Nijenhuis coordinates and symplectic connections.

Generalization of bi-Hamiltonian systems in (3+1) dimension, possessing partner symmetries

We study bi-Hamiltonian structure of a general equation which possesses partner symmetries. The general form of such second-order PDEs with four independent variables was determined in the paper Sheftel and Malykh (2009) on a classification of second-order PDEs which have this property. We apply Dirac’s theory of constraints to this general equation. We formulate the equation in a two-component form and present the Lax pair of Olver–Ibragimov–Shabat type. Under some constraints imposed on constant coefficients of this equation, we obtain its bi-Hamiltonian structure. Therefore, by Magri’s theorem it is a completely integrable bi-Hamiltonian system in ( 3 + 1 ) dimensions. We also showed that with suitable choices of constant coefficients the equation is reduced to the well known integrable bi-Hamiltonian systems in ( 3 + 1 ) dimension.

On integrability of some bi-Hamiltonian two field systems of partial differential equations

We continue the study of integrability of bi-Hamiltonian systems with a compatible pair of local Poisson structures (H0, H1), where H0 is a strongly skew-adjoint operator. This is applied to the construction of some new two field integrable systems of PDE by taking the pair (H0, H1) in the family of compatible Poisson structures that arose in the study of cohomology of moduli spaces of curves.