Department of Mathematics,

University of Ljubljana,

Jadranska 19,

1000 Ljubljana

SLOVENIA

E-mail: pavle.saksida@fmf.uni-lj.si, saksida@crm.umontreal.ca

**Maxwell-Bloch equations and chains of C. Neumann oscillators**

**Abstract:**

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.