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 diffierential 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.