Joint work with M. Agrotis, S.A. Glasgow, and J.V. Moloney.

The Reduced Maxwell-Bloch (RMB) Equations are a semiclassical description of the material response of a two-level system to a classical electromagnetic field. The RMB equations have some similarity to the integrable self-induced transparency (SIT) equations which describe the envelope of the carrier wave. We show that the RMB equations are completely integrable by constructing a Lax pair which generates them (as well as generalizations which include the presence of a permanent dipole).

We go on to describe the geometry of the underlying loop algebra for these systems and the associated Bäcklund transformations. Physically, the RMB model is quite different from SIT in that it gives a description of the electric field rather than just its envelope. Recent technological advances have made possible the creation of ultrafast laser pulses (existing on femtosecond time scales) and lasting only half a wavelength (so-called) half cycle pulses). The RMB equations have been proposed as a model for such optical devices. In particular, the solutions we construct by Bäcklund transformations enable one to describe the effects of inhomogeneous broadening in the RMB model.

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