Poisson vertex algebra and integrable Hamiltonian PDE

Abstract

This paper explains the relation between Lie conformal algebras, vertex algebras, and Poisson vertex algerbas, which is similar to the relation between Lie algebras, associative algebras, and Poisson algebras. Also we explain how Poisson vertex algebra structure is related to integrable Hamiltonian PDE. There is a significant theorem determining whether a Hamiltonian PDE is integrable or not, called Lenard scheme. Through this, we show that KdV equation is integrable and find infinitely many conserved densities. At the end, this paper introduces short introduction to classical affine W-algebras.