Department of Mathematics,
University of Ljubljana,
Jadranska 19,
1000 Ljubljana

Maxwell-Bloch equations and chains of C. Neumann oscillators

The Maxwell-Bloch equations are an integrable system of PDE's which, roughly speaking, serve as a model for a laser. I will consider a discretization of this system with respect to the spatial variable, leaving the time continuous. We obtain a system of ODE's which describes a chain of globally interacting C. Neumann oscillators on three-sphere. In this context I shall describe a method which yields a full system of integrals of motion of this C. Neumann chain. This new method of constructing converved quantities is a modification of the zero-curvature condition. The existence of the integrals stems from the fact that a certain curvature takes values in a suitable subalgebra of the Lie algebra of the structure group, but this curvature is in general non-zero. I shall also mention a new Hamiltonian structure of the
Maxwell-Bloch equations.