Solitary-Waves in Self-Induced Transparency
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 eigenvalues1,2. 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 work4,5 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 transform6,7. Also, to all orders in perturbation theory, there is a continuum of solitary-wave solutions.
KeywordsSingular Perturbation Electric Field Amplitude Pulse Solution Exact Integrability Solitary Pulse
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