Poisson vertex algebras in the theory of Hamiltonian equations

Understand

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.