Inverse scattering problems for first - order systems

In this thesis, the Zakharov-Shabat scattering problem and several types of Landau-Lifschitz scattering problems are considered. The inverse scattering problem is that of seeking one or more coefficients in a system of differential equation from the scattering data which, generally, consists of a reflection coefficient and bound state data. We assume mainly that the coefficients to be determined are of half line support, i.e. they are equal to zero on a half line. Such cases can arise on natural physical grounds, and they can be very good approximations in case of reasonably rapid decay. Assuming that the coefficients have half line support, we present the uniqueness of the inverse scattering problems as well as develop several efficient numerical algorithms to reconstruct the coefficients via a time domain approach. Also, a relation of the Zakharov-Shabat scattering problem and the Landau-Lifschitz scattering problem is investigated. Some exact theory for the inverse scattering problem with no support restriction is developed by means of corresponding half line support problems.