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arXiv · 1806.05536

Poisson $λ$-brackets for differential-difference equations

Alberto De Sole·Victor G. Kac·Daniele Valeri·Minoru Wakimoto
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Abstract

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

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BibTeXRIS

Alberto De Sole, Victor G. Kac, Daniele Valeri, Minoru Wakimoto. 2018-06-18. Poisson $λ$-brackets for differential-difference equations. https://arxiv.org/abs/1806.05536

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