Poisson vertex algebras and Hamiltonian PDE

Abstract

We discuss the theory of Poisson vertex algebras and their generalizations in relation to integrability of Hamiltonian PDE. In particular, we discuss the theory of affine classical W-algebras and apply it to construct a large class of integrable Hamiltonian PDE. We also discuss non commutative Hamiltonian PDE in the framework of double PVA, and differential-difference Hamiltonian equations in the framework of multiplicative PVA.

This article is the preprint version of our contribution to the book titled "Encyclopedia of Mathematical Physics 2nd edition". v2: minor editing and corrections following the referee's suggestions