Properties of an Alternate Lax Description of the KdV Hierarchy

TL;DR

This paper systematically explores an alternative Lax operator formulation of the KdV hierarchy, revealing natural recursion relations and Hamiltonian structures, and providing a unified geometric perspective.

Contribution

It introduces a new Lax description using the geometrical recursion operator, clarifies the Hamiltonian structures, and simplifies the understanding of conserved quantities in the KdV hierarchy.

Findings

Formulated the Lax equation for the n-th flow.

Constructed Hamiltonians for commuting flows.

Provided a unified definition of Hamiltonian structures.

Abstract

We study systematically the Lax description of the KdV hierarchy in terms of an operator which is the geometrical recursion operator. We formulate the Lax equation for the $n$-th flow, construct the Hamiltonians which lead to commuting flows. In this formulation, the recursion relation between the conserved quantities follows naturally. We give a simple and compact definition of all the Hamiltonian structures of the theory which are related through a power law.