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.
Poisson Λ-brackets for Differential-Difference Equations / De Sole, A.; Kac, V. G.; Valeri, D.; Wakimoto, M.. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2020:13(2020), pp. 4144-4190. [10.1093/imrn/rny242]
Poisson Λ-brackets for Differential-Difference Equations
De Sole A.;Valeri D.;
2020
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 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.| File | Dimensione | Formato | |
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