Poisson vertex algebras in the theory of Hamiltonian equations
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Kac Poisson vertex algebras....pdf
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Author(s)
Barakat, Aliaa
De Sole, Alberto
Kac, Victor
Date Issued
December 2009
Journal
Japanese Journal of Mathematics
Publisher
Springer Japan
Citation
Barakat, Aliaa, Alberto De Sole, and Victor Kac. “Poisson Vertex Algebras in the Theory of Hamiltonian Equations.” Japanese Journal of Mathematics 4.2 (2009) : 141-252-252.
Version
Author's final manuscript
Abstract
We lay down the foundations of the theory of Poisson vertex algebras aimed at its
applications to integrability of Hamiltonian partial differential equations. Such an equation
is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian
equations, which admit an infinite sequence of linearly independent integrals of motion in
involution. The construction of a hierarchy and its integrals of motion is achieved by making
use of the so called Lenard scheme. We find simple conditions which guarantee that the
scheme produces an infinite sequence of closed 1-forms ωj , j ∈ Z+, of the variational complex Ω. If these forms are exact, i.e. ωj are variational derivatives of R some local functionals
hj, then the latter are integrals of motion in involution of the hierarchy formed by the
corresponding Hamiltonian vector fields. We show that the complex Ω
is exact, provided
that the algebra of functions V is “normal”; in particular, for arbitrary V, any closed form
in Ω
becomes exact if we add to V a finite number of antiderivatives. We demonstrate
on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works.
We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type.
Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures,
and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11537-009-0932-y
