Poisson Λ-brackets for Differential–Difference Equations

Abstract We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential–difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian partial differential equations (PDE). We classify multiplicative Poisson $\lambda$-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 $\lambda$-brackets of order 1 and 2, to establish integrability of the Volterra chain.