Integrable bi-Hamiltonian Systems Research Articles

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}}.

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.

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.

Bi-Hamiltonian structure of an asymmetric heavenly equation

In the paper of Sheftel and Malykh (2009 J. Phys. A: Math. Theor. 42 395202) on the classification of second-order PDEs with four independent variables that possess partner symmetries, an asymmetric heavenly equation appears as one of the canonical equations admitting partner symmetries. Here, for the asymmetric heavenly equation formulated in a two-component form, we present the Lax pair of Olver–Ibragimov–Shabat type and obtain its multi-Hamiltonian structure. Therefore, by Magri’s theorem, it is a completely integrable bi-Hamiltonian system in four dimensions.

The generalized Kupershmidt deformation for the fifth-order coupled KdV equations hierarchy

Based on the Kupershmidt deformation, we propose the generalized Kupershmidt deformation (GKD) to construct new systems from integrable bi-Hamiltonian system. As applications, the generalized Kupershmidt deformation of the fifth-order coupled KdV equations hierarchy with self-consistent sources and its Lax representation are presented.

The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems

Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented by Kupershmidt [Phys. Lett. A 372, 2634 (2008)], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: Korteweg–de Vries (KdV) equation, Boussinesq equation, Jaulent–Miodek equation, and Camassa–Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a t-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.

SYSTEMS OF HESS–APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY

We start with a review of a class of systems with invariant relations, so called systems of Hess–Appel'rot type that generalizes the classical Hess–Appel'rot rigid body case. The systems of Hess–Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess–Appel'rot type carry an interesting combination of both integrable and non-integrable properties.Further, following integrable line, we study partial reductions and systems having what we call the Zhukovskii property: These are Hamiltonian systems on a symplectic manifold M with actions of two groups G and K; the systems are assumed to be K-invariant and to have invariant relation Φ = 0 given by the momentum mapping of the G-action, admitting two types of reductions, a reduction to the Poisson manifold P = M/K and a partial reduction to the symplectic manifold N0= Φ-1(0)/G; final and crucial assumption is that the partially reduced system to N0is completely integrable. We prove that the Zhukovskii property is a quite general characteristic of systems of Hess–Appel'rot type. The partial reduction neglects the most interesting and challenging part of the dynamics of the systems of Hess–Appel'rot type — the non-integrable part, some analysis of which may be seen as a reconstruction problem.We show that an integrable system, the magnetic pendulum on the oriented Grassmannian Gr+(n, 2) has a natural interpretation within Zhukovskii property and that it is equivalent to a partial reduction of certain system of Hess–Appel'rot type. We perform a classical and algebro-geometric integration of the system in dimension four, as an example of a known isoholomorphic system — the Lagrange bitop.The paper presents a lot of examples of systems of Hess–Appel'rot type, giving an additional argument in favor of further study of this class of systems.

Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras

Recently Hirota and Kimura presented a new discretization of the Euler top with several remarkable properties. In particular this discretization shares with the original continuous system the feature that it is an algebraically completely integrable bi-Hamiltonian system in three dimensions. The Hirota-Kimura discretization scheme turns out to be equivalent to an approach to numerical integration of quadratic vector fields that was introduced by Kahan, who applied it to the two-dimensional Lotka-Volterra system. The Euler top is naturally written in terms of the $\mathfrak{so}(3)$ Lie-Poisson algebra. Here we consider algebraically integrable systems that are associated with pairs of Lie-Poisson algebras in three dimensions, as presented by Gumral and Nutku, and construct birational maps that discretize them according to the scheme of Kahan and Hirota-Kimura. We show that the maps thus obtained are also bi-Hamiltonian, with pairs of compatible Poisson brackets that are one-parameter deformations of the original Lie-Poisson algebras, and hence they are completely integrable. For comparison, we also present analogous discretizations for three bi-Hamiltonian systems that have a transcendental invariant, and finally we analyze all of the maps obtained from the viewpoint of Halburd's Diophantine integrability criterion.

Algebraic properties of Gardner’s deformations for integrable systems

We formulate an algebraic definition of Gardner’s deformations for completely integrable bi-Hamiltonian evolutionary systems. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and hyperbolic Liouville-type systems. We find an exactly solvable two-component extension of the Liouville equation.

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.

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.

Kronecker webs, bihamiltonian structures, and the method of argument translation

We show that manifolds which parameterize values of first integrals of integrable finite-dimensional bihamiltonian systems carry a geometric structure which we call aKronecker web. We describe two opposite direction functors between Kronecker webs and integrable bihamiltonian structures: one is left inverse to the other. Conjecturally, these two functors are mutually inverse (for “small” open subsets of the manifolds in question).

Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons

We investigate how the non-analytic solitary wave solutions -- peakons and compactons -- of an integrable bi-Hamiltonian system arising in fluid mechanics, can be recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for the reduced dynamical system. This phenomenon is examined to understand the important effect of linear dispersion terms on the analyticity of such homoclinic orbits.

Completely integrable bi-Hamiltonian systems

We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that under some natural hypothesis, such a structure exists in a neighborhood of an invariant torus if, and only if, the graph of the Hamiltonian function is a hypersurface of translation, relative to the affine structure determined by the action variables. This generalizes a result of Brouzet for dimension four.