Date of Award

8-1-2012

Document Type

Campus Access Thesis

Degree Name

Master of Science (MS)

Department

Physics, Applied

First Advisor

Maxim Olchanyi

Second Advisor

Stephen Arnason

Third Advisor

Bala Sundaram

Abstract

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.

Comments

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