Solitary-Waves in Self-Induced Transparency

Abstract

Numerous systems are known to give rise to velocity or wavelength “selection”: crystal growth, fluid flow, shock waves, etc... Mathematically, the velocity or wavelength appear as non-linear eigenvalues^1,2. It is often the case that the “unperturbed” problem (e.g., the zero-surface tension limit for crystal growth and fluid flow) has a continuous non-linear eigenvalue spectrum because of some underlying symmetry. The addition of a perturbation will usually break this symmetry and will lead to a discrete spectrum, that is to “selection”. Exactly integrable PDEs provide a particularly interesting ground for selection studies. They typically have a continuum of soliton solutions with different velocities or amplitudes. Adding a generic perturbation destroys their exact integrability. Solitons should disappear or become solitary-waves under such perturbations. There are many examples where ordinary perturbations destroy the family of solitons, leaving a single solitary-wave³. The effects of singular perturbations are more subtle, but have been investigated in the last few years for the cases of the standard exactly integrable PDEs (KdV, NLS, SG; see several of the articles in these proceedings). The conclusion of these works is that higher derivatives destroy all solitary-waves; generally, steady-state solutions have capillary waves going out all the way to infinity from the main peak. The purpose of this article is to report on some work^4,5 on the coupled Maxwell-Bloch equations. In the slowly varying envelope approximation (SVEA), these PDEs reduce to an exactly integrable system which can be treated by the inverse scattering transform^6,7. Also, to all orders in perturbation theory, there is a continuum of solitary-wave solutions.