Date of Award
Campus Access Thesis
Master of Science (MS)
In this thesis we will explore some interconnections among the following topics: 1. Reﬂectionless potentials; 2. inverse problems in physics; and 3. the solitonic solutions of completely integrable nonlinear partial differential equations. We will begin by discussing the form of the reﬂectionless potential and the nature of solutions of the associated Schrödinger equation. Then we will give an overview of the inverse problem in physics, before discussing the inverse scattering problem and its solutions as formulated by Gel’fand, Levitan, and Marchenko. The construction of reﬂectionless potential appears naturally in this setting. Lastly, we will explore how the inverse scattering problem enters into the scattering transform, which is a method of constructing exact solutions for nonlinear partial differential equations. A notable type of these solutions is one which consists of a ﬁnite number of solitons. It will turn out that such solutions are connected with the appearance of the reﬂectionless potential in the associated scattering problem.
Leonard, Louis Lesly, "Inverse Scattering Problems, Reflectionless Potentials, and Solitons" (2012). Graduate Masters Theses. 139.