Poisson $\lambda$-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 PDE. We classify multiplicative Poisson $\lambda$-brackets in one difference variable up to order 5. Applying the Lenard-Magri scheme to a compatible pair of multiplicative Poisson $\lambda$-brackets of order 1 and 2, we establish integrability of some differential-difference equations, generalizing the Volterra chain.