A Lax Equation for the Non-Linear Sigma Model

TL;DR

This paper introduces a Lax equation for the non-linear sigma model, revealing its bi-Hamiltonian structure, infinite conserved charges, and establishing its complete integrability through explicit recursion operators.

Contribution

It presents the first formulation of a Lax equation for the non-linear sigma model, demonstrating its bi-Hamiltonian nature and complete integrability with two sets of conserved charges.

Findings

Two infinite sets of conserved charges identified

The system exhibits bi-Hamiltonian structure with compatible Hamiltonian forms

Conserved charges are in involution, confirming integrability

Abstract

We propose a Lax equation for the non-linear sigma model which leads directly to the conserved local charges of the system. We show that the system has two infinite sets of such conserved charges following from the Lax equation, much like dispersionless systems. We show that the system has two Hamiltonian structures which are compatible so that it is truly a bi-Hamiltonian system. However, the two Hamiltonian structures act on the two distinct sets of charges to give the dynamical equations, which is quite distinct from the behavior in conventional integrable systems. We construct two recursion operators which connect the conserved charges within a given set as well as between the two sets. We show explicitly that the conserved charges are in involution with respect to either of the Hamiltonian structures thereby proving complete integrability of the system. Various other interesting features are also discussed.