Complete asymptotic representation of an electromagnetic pulse in a long two-level amplifier

The propagation of an electromagnetic wave in a nonlinear two-level medium described in the framework of Lamb's semiclassical theory is considered. The corresponding system of Maxwell-Bloch equations is investigated by the inverse scattering method with a view to constructing a complete asymptotic expansion of its solutions at large separation from the edge of the region. In the neighborhood of the wave front, the solution is described by a Painlevé equation, whereas far from the front the solution goes over to a rapidly oscillating self-similar regime. In the intermediate region, the parameters of these asymptotic solutions are matched by comparing the corresponding scattering data.