Japanese Journal of Mathematics

, Volume 8, Issue 2, pp 233–347 | Cite as

Non-local Poisson structures and applications to the theory of integrable systems

  • Alberto De SoleEmail author
  • Victor G. Kac


We develop a rigorous theory of non-local Poisson structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard–Magri scheme of integrability to a pair of compatible non-local Poisson structures. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as hyperbolic equations.

Keywords and phrases

non-local Poisson vertex algebra non-local Poisson structure rational matrix pseudo-differential operators Lenard–Magri scheme of integrability bi-Hamiltonian integrable hierarchies 

Mathematics Subject Classification (2010)

37K10 (primary) 35Q53 17B80 17B69 37K30 17B63 (secondary) 


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Copyright information

© The Mathematical Society of Japan and Springer Japan 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Department of MathematicsMIT.CambridgeUSA

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