On a poisson reduction for Gel'fand-Zakharevich manifolds

We formulate and discuss a reduction theorem for Poisson pencils associated with a class of integrable systems, defined on bi-Hamiltonian manifolds, recently studied by Gel'fand and Zakharevich. The reduction procedure is suggested by the bi-Hamiltonian approach to the separation of variables problem.     

Drinfeld-Sokolov reduction on a simple lie algebra from the bihamiltonian point of view

We show that the Drinfeld-Sokolov reduction can be framed in the general theory of bihamiltonian manifolds, with the help of a specialized version of a reduction theorem for Poisson manifolds by Marsden and Ratiu.This work has been supported by the Italian MURST and by the GNFM of the Italian C.N.R.     

On bi-Hamiltonian structure of two-component Novikov equation

In this Letter, we present a bi-Hamiltonian structure for the two-component Novikov equation. We also show that proper reduction of this bi-Hamiltonian structure leads to the Hamiltonian operators found by Hone and Wang for the Novikov equation.     

A Three-component Extension of the Camassa–Holm Hierarchy

We introduce a bi-Hamiltonian hierarchy on the loop-algebra of endowed with a suitable Poisson pair. It gives rise to the usual Camassa–Holm (CH) hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a three-component extension of the CH equation.     

On classification and construction of algebraic Frobenius manifolds

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld–Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.     

Bi-Hamiltonian structure in Frenet-Serret frame

We reduce the problem of constructing bi-Hamiltonian structure in three dimensions to the solutions of a Riccati equation in moving coordinates of Frenet-Serret frame. All explicitly constructed examples in the literature are exhausted by constant solutions. We conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi-Hamiltonian.     

Bi-Hamiltonian systems of deformation type

In this paper, after some recalls about Poisson cohomology, we first study what the general method is in order to obtain a bi-Hamiltonian formulation of a given Hamiltonian system by means of a deformation. Then we show that the bi-Hamiltonian formulation which results from the deformation of a Poisson structure by means of a suitable non-Noether symmetry cannot explain the complete integrability for a large class of Arnold–Liouville integrable systems; next we prove that the deformation must be made in this context by a suitable mastersymmetry. At last, we give several examples.     

Comment: On a recently proposed relation between oHS and Ito systems

The bi-Hamiltonian structure of original Hirota–Satsuma system proposed by Roy [Phys. Lett. A 249 (1999) 55] based on a modification of the bi-Hamiltonian structure of Ito system is incorrect.     

Compatible Lie algebroids and the periodic Toda lattice

We give the set of maps from to the structure of a Poisson manifold endowed with a pair of compatible Lie algebroids. A suitable reduction process, of the Marsden–Ratiu type, yields a smaller manifold with the same geometrical properties as the original manifold. Moreover, is a bi-Hamiltonian manifold and the flows naturally defined on it are the periodic Toda flows.     

Toda Equations, bi-Hamiltonian Systems, and Compatible Lie Algebroids

We present the bi-Hamiltonian structure of Toda₃, a dynamical system studied by Kupershmidt as a restriction of the discrete KP hierarchy. We derive this structure by a suitable reduction of the set of maps from Z _d to GL(3,R), in the framework of Lie algebroids.     

Recursion operators and bi-Hamiltonian structures in multidimensions. I

The algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied. In particular, the factorization of certain operators, called recursion operators, establishes the bi-Hamiltonian nature of all these equations. Recently, we have presented the recursion operator and the bi-Hamiltonian formulation of the Kadomtsev-Petviashvili equation, a two spatial dimensional analogue of the Korteweg-deVries equation. Here we present the general theory associated with recursion operators for bi-Hamiltonian equations in two spatial and one temporal dimensions. As an application we show that general classes of equations, which include the Kadomtsev-Petviashvili and the Davey-Stewartson equations, possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct Poisson brackets. Furthermore, we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems.     

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the non-commutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the bi-Hamiltonian structure and noncommutative conserved densities of the extended noncommutative Toda hierarchy by means of the R-matrix formalism. This extended noncommutative Toda hierarchy can be reduced to the extended multicomponent Toda hierarchy, extended Z_N?-Toda hierarchy, extended Toda hierarchy respectively by reductions on Lie algebras.     

Integrable Couplings of the Generalized AKNS Hierarchy with an Arbitrary Function and Its Bi-Hamiltonian Structure

We construct a new loop algebra \(\widetilde{A_{3}}\), which is used to set up an isospectral problem. Then a new integrable couplings of the generalized AKNS hierarchy is derived, which possesses bi-Hamiltonian structure and contains an arbitrary spatial function. As its reduction, we gain the integrable couplings of the Schrödinger equation. Furthermore, many conserved quantities of the integrable couplings are obtained.     

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.     

Singularities of Bi-Hamiltonian Systems

We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types.     

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.     

Bi-Hamiltonian Structure of a Three-Component Camassa-Holm Type Equation

A recently proposed three-component Camassa-Holm equation is considered. It is shown that this system is a bi-Hamiltonian system.     

A cohomological obstruction for global quasi-bi-Hamiltonian fields

We introduce the notion of integrating factor for a 1-form which is an inner product of a vector fields and a 2-form, and the notion of weakly bi-Hamiltonian field also, which is locally quasi-bi-Hamiltonian. A cohomological class in some first cohomology space is associated to such vector fields when this is weakly bi-Hamiltonian and defined relatively to the above 1-form. This class is a cohomological obstruction to the existence of a global integrating factor for the 1-form.