On third Poisson structure of KdV equation

TL;DR

This paper explores the third Poisson structure of the KdV equation, demonstrating its diagonalization in certain variables and proposing a quantum path integral formulation based on this structure.

Contribution

It introduces a new representation of the third Poisson structure of KdV in Lagrange variables and develops a quantum path integral approach.

Findings

Diagonalization of the third Poisson structure in Lagrange variables

Connection to reduced WZNW model and 2D gravity variables

Proposal of a quantum path integral formulation for KdV

Abstract

The third Poisson structure of KdV equation in terms of canonical ``free fields'' and reduced WZNW model is discussed. We prove that it is ``diagonalized'' in the Lagrange variables which were used before in formulation of 2D gravity. We propose a quantum path integral for KdV equation based on this representation.