# PreHamiltonian and Hamiltonian operators for differential-difference equations

Carpentier, Sylvain, Mikhailov, Alexander V., Wang, Jing Ping (2020) PreHamiltonian and Hamiltonian operators for differential-difference equations. Nonlinearity, 33 (3). ISSN 0951-7715. E-ISSN 1361-6544. (doi:10.1088/1361-6544/ab5912) (Access to this publication is currently restricted. You may be able to access a copy if URLs are provided) (KAR id:78759)

## 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. The definition of a rational Hamiltonian operator can be reformulated in terms of its factors which simplifies the theory and makes it useful for applications. In particular we show that for a given rational Hamiltonian operator H in order to find a second Hamiltonian operator K compatible with H one only needs to find a preHamiltonian pair A and B such that K = AB−1H is skew-symmetric. We apply our theory to study multi Hamiltonian structures of Narita-Itoh-Bogayavlensky and AdlerPostnikov equations.

Item Type: Article 10.1088/1361-6544/ab5912 Q Science > QA Mathematics (inc Computing science) Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Jing Ping Wang 15 Nov 2019 16:28 UTC 06 Feb 2020 04:20 UTC https://kar.kent.ac.uk/id/eprint/78759 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-6874-5629