Poisson vertex algebras in the theory of Hamiltonian equations

TL;DR

This paper develops the theory of Poisson vertex algebras to analyze the integrability of Hamiltonian PDEs, introducing new conditions for the Lenard scheme to generate infinite hierarchies of conserved quantities.

Contribution

It establishes foundational conditions for Poisson vertex algebras in Hamiltonian equations, extends the Lenard scheme to Dirac structures, and discovers a new integrable hierarchy.

Findings

Proved the exactness of the variational complex under certain conditions.

Extended the Lenard scheme to Dirac structures.

Discovered a new integrable hierarchy, CNW of HD type.

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 \omega_j, j in Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j are variational derivatives of some local functionals \int h_j, then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex \Omega is exact, provided that the algebra of functions V is "normal"; in particular, for arbitrary V, any closed form in \Omega becomes exact if we add to V a finite number of antiderivatives. We demonstrate on the examples of 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.