Poisson λ-brackets for Differential–Difference Equations

Abstract

© 2018 The Author(s). Published by Oxford University Press. All rights reserved. We introduce the notion of a multiplicative Poisson λ-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson λ-bracket plays in the theory of Hamiltonian partial differential equations (PDE). We classify multiplicative Poisson λ-brackets in one difference variable up to order 5. As an example, we demonstrate how to apply the Lenard-Magri scheme to a compatible pair of multiplicative Poisson λ-brackets of order 1 and 2, to establish integrability of the Volterra chain.