## Graduation Semester and Year

2018

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Tuncay Aktosun

## Abstract

A first-order system of two linear ordinary differential equations is analyzed. The linear system contains a spectral parameter, and it has two coefficients that are functions of the spatial variable ��. Those two functions act as potentials in the linear system and they also linearly contain the spectral parameter λ, and hence they are referred to as energy-dependent potentials. Such a linear system arises in the solution to a pair of integrable nonlinear partial differential equations (known as the derivative nonlinear Schrödinger equations) via the so-called inverse scattering transform method. The direct and inverse problems for the corresponding first-order linear system with energy-dependent potentials are investigated. In the direct problem, when the two potentials belong to the Schwartz class, the properties of the corresponding scattering coefficients and so-called bound-state data are derived. In the inverse problem, the two potentials are recovered from the scattering data set consisting of the scattering coefficients and bound-state data. The solutions to the direct and inverse problems are achieved by relating the scattering data and the potentials in the energy-dependent system to those in a pair of first-order system with energy-independent potentials. An alternate solution to the inverse problem is given by formulating a linear integral equation (referred to as the alternate Marchenko integral equation), and the energy-dependent potentials are recovered with the help of the solution to the alternate Marchenko equation.

## Keywords

Scattering, Inverse scattering, Differential equation

## Disciplines

Mathematics | Physical Sciences and Mathematics

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Ercan, Ramazan, "Scattering and Inverse Scattering on the Line for a First-Order System with Energy-Dependent Potentials" (2018). *Mathematics Dissertations*. 112.

https://mavmatrix.uta.edu/math_dissertations/112

## Comments

Degree granted by The University of Texas at Arlington