Japanese Journal of Mathematics

, Volume 8, Issue 2, pp 233–347

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


DOI: 10.1007/s11537-013-1306-z

Cite this article as:
De Sole, A. & Kac, V.G. Jpn. J. Math. (2013) 8: 233. doi:10.1007/s11537-013-1306-z


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) 

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