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

Abstract

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.

120 pages. This paper is an edited version of the merger of the two papers with archive numbers arXiv:1210.1688 and arXiv:1211.2391. Version 2: minor corrections. Added keywords and MSC codes. To appear in the Japanese Journal of Mathematics

References in corpus (2)

• The short pulse equation is integrable
• Some remarks on non-commutative principal ideal rings

Cited by in corpus (23)

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• Dirac reduction for Poisson vertex algebras
• Classical W-algebras and generalized Drinfeld-Sokolov hierarchies for minimal and short nilpotents
• A new scheme of integrability for (bi)Hamiltonian PDE
• On deformations of multidimensional Poisson brackets of hydrodynamic type
• Three computational approaches to weakly nonlocal Poisson brackets
• On a class of third-order nonlocal Hamiltonian operators
• Some remarks on non-commutative principal ideal rings
• Local and non-local multiplicative Poisson vertex algebras and differential-difference equations
• A new approach to the Lenard-Magri scheme of integrability
• Weakly nonlocal Poisson brackets: tools, examples, computations
• Integrability of Dirac reduced bi-Hamiltonian equations
• Singular degree of a rational matrix pseudodifferential operator
• Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds
• On integrability of some bi-Hamiltonian two field systems of PDE
• Hamiltonian structures for integrable hierarchies of Lagrangian PDEs
• Double Multiplicative Poisson Vertex Algebras
• Two-component integrable generalizations of Burgers equations with nondiagonal linearity
• Dirac reductions and Classical W-algebras
• Some Remarks on the Operators' Formalism for Nonlocal Poisson Brackets