Carpentier, S, Mikhailov, AV and Wang, JP (2020) PreHamiltonian and Hamiltonian operators for differential-difference equations. Nonlinearity, 33 (3). 915. ISSN 0951-7715
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 , 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 . 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 −1 H is skew-symmetric. We apply our theory to study multi-Hamiltonian structures of Narita–Itoh–Bogayavlensky and Adler–Postnikov equations.
Metadata
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Copyright, Publisher and Additional Information: | © 2020 IOP Publishing Ltd & London Mathematical SocietyThis is an author produced version of an article published in Nonlinearity. Uploaded in accordance with the publisher's self-archiving policy. | ||||
Keywords: | difference equations; Hamiltonian operetors; bi-Hamiltonian structure; pseudo-difference operators; integrable systems; difference algebra; preHamiltonian operators | ||||
Dates: |
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Institution: | The University of Leeds | ||||
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) | ||||
Funding Information: |
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Depositing User: | Symplectic Publications | ||||
Date Deposited: | 23 Oct 2019 13:34 | ||||
Last Modified: | 23 Jan 2021 01:38 | ||||
Status: | Published | ||||
Publisher: | London Mathematical Society | ||||
Identification Number: | https://doi.org/10.1088/1361-6544/ab5912 |