Interaction of radiation with matter: Integrable problems

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Abstract
We construct an extension of the spectral transform theory that allows us to build nonlinear systems of coupled waves that are integrable for arbitrary boundary values. The related time evolution of the spectral transform is in general nonlinear. This result has many important applications in physics, and we apply the procedure to plasma waves (laser-plasma interaction), to quantum electronics [self-induced transparency (SIT) and laser-pulse amplification], and to nonlinear optics (stimulated Raman scattering). In the case of laser-plasma interaction, we obtain an exact model for the description of the total reflexivity due to stimulated Brillouin scattering. For SIT, we show that the presence in the initial state of the medium of some atoms in an excited state drastically modifies the related time evolution of the spectral data, possibly making the problem unsolvable. In the case of laser-pulse amplification, we prove that the presence of background noise in the firing laser pulse drastically modifies the long-distance behavior of the solitons. Finally, for the general process of stimulated Raman scattering, we give the correct evolution of the spectral transform and show that the Stokes wave becomes rapidly totally localized; in other words, the system naturally evolves into a pure soliton state, whatever may be the initial profile of the acoustic wave. In the same context, another quite interesting example is studied: it is the first instance of an integrable system which develops a singularity in a finite time (for which the solution blows up). The physical application is under study.