## Poisson vertex algebras in the theory of Hamiltonian equations

##### Author(s)

Barakat, Aliaa; De Sole, Alberto; Kac, Victor
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Show full item record##### 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.

##### Date issued

2009-12##### Department

Massachusetts Institute of Technology. Department of Mathematics##### 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

##### ISSN

0289-2316

1861-3624