PreHamiltonian and Hamiltonian operators for differential-difference equations

Abstract

In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo-difference Hamiltonian operator can be represented as a ratio AB −1 of two difference operators with coefficients from a difference field F where A is preHamiltonian. A difference operator A is called preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields on F. We show that a skew-symmetric difference operator is Hamiltonian if and only if it is preHamiltonian and satisfies simply verifiable conditions on its coefficients. We show that if H is a rational Hamiltonian operator, then to find a second Hamiltonian operator K compatible with H is the same as to find a preHamiltonian pair A and B such that AB −1 H is skew-symmetric. We apply our theory to non-trivial multi-Hamiltonian structures of Narita-Itoh-Bogayavlensky and Adler-Postnikov equations.

Hamiltonian and recursion operators for a discrete analogue of the Kaup-Kupershmidt equation

“…Roughly speaking, a recursion operator is a linear operator R mapping a symmetry to a new symmetry. For an evolutionary equation (12), it satisfies…”

“…Equation ( 7) is Hamiltonian with respect to it. To prove that H is Hamiltonian in theorem 4.1, we apply the formalism of rational and pre-Hamiltonian operators described in [10,12].…”

Multi-component Toda lattice in centro-affine $${\mathbb R}^n$$

“…(2) Another method to prove this statement is using recent results on preHamiltonian operators [19,20]. We call a difference operator preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields.…”

A Darboux–Getzler Theorem for Scalar Difference Hamiltonian Operators

“…All the results obtained in this paper apply to local Hamiltonian operators, namely to operators which are polynomials in S and S −1 . A theory of rational Hamiltonian operator has been recently developed [5]; the Poisson cohomology of such a larger class of structures is the natural further topic in the direction of their classification.…”

“…The main purpose of this paper is the extension to the difference case of the notion of Poisson cohomology of a Hamiltonian structure. In this context, Hamiltonian structures are given by difference operators (we call them, in analogy with the differential case, local operators), or ratios of difference operators [5]. Our principal result is the computation of the Poisson cohomology for a scalar, order (−1, 1) difference Hamiltonian operator.…”