Graduation Semester and Year
Summer 2024
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
Dr. Tuncay Aktosun
Second Advisor
Dr. Hristo Kojouharov
Third Advisor
Dr. Yue Liu
Fourth Advisor
Dr. Souvik Roy
Abstract
We consider the full-line direct and inverse scattering problems for the third-order ordinary differential equation containing two potentials decaying sufficiently fast at infinity. The direct scattering problem consists of the determination of the scattering data set when the two potentials are known. The scattering data set is made up of the corresponding scattering coefficients and the bound-state information. On the other hand, the inverse scattering problem involves the recovery of the two potentials when the scattering data set is available. We formulate the inverse scattering problem via a related Riemann--Hilbert problem on the complex plane. We describe the recovery of the two potentials from the solution to that Riemann--Hilbert problem. We also mention how the Riemann--Hilbert problem leads to a system of Marchenko integral equations. The recovery of the potentials from the solution to the Marchenko system will be published elsewhere.
Keywords
direct, inverse, scattering, theory, Riemann-Hilbert, Marchenko
Disciplines
Other Mathematics
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Toledo, Ivan, "The direct and inverse scattering problems for the third-order operator" (2024). Mathematics Dissertations. 163.
https://mavmatrix.uta.edu/math_dissertations/163