A Lax Equation for the Non-Linear Sigma Model

Details A Lax Equation for the Non-Linear Sigma Model A Lax Equation for the Non-Linear Sigma Model

2002-08-23

arxiv.org/abs/hep-th/0208172v1

Phys.Lett.B546:167-176,2002

Physics

High Energy Physics

High Energy Physics - Theory

Latex, 13 pages

Scientific paper

10.1016/S0370-2693(02)02663-1

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.

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