Committee Chair

Cox, Christopher

Committee Member

Belinskiy, Boris; Nichols, Roger; Kong, Lingju

Department

Dept. of Mathematics

College

College of Engineering and Computer Science

Publisher

University of Tennessee at Chattanooga

Place of Publication

Chattanooga (Tenn.)

Abstract

This dissertation explores various solutions to nonlinear reaction-diffusion systems, focusing primarily on the nonlinear Schrödinger equation. Three types of solutions are investigated: radial solutions (with no angular dependence), vortex solutions (with angular dependence but no radial phase dependence), and spiral solutions (which have radial phase dependence). The study considers free particles without potential and trapped particles inside the cylindrical potential with an impenetrable barrier. We show that spiral solutions to the nonlinear Schrödinger equation only exist for a radially constant phase by considering a related system of nonlinear ordinary differential equations. This system is shown to be a special case of the $\lambda$-$\omega$ reaction-diffusion system, whose spiral solutions exist for a nonconstant phase.

Acknowledgments

I would like to thank the Department of Mathematics and the Graduate School at UTC for allowing me to obtain my Ph.D. I would like to acknowledge support from the Departments of Mathematics and Computer Science, as well as the Quantum Initiative for my assistantship funding. I would like to thank Dr. Rubenstein for his discoveries, and willingness to share his research with me and the faculty at UTC. I would also like to thank Dr. Cox and Dr. Belinskiy for their invaluable knowledge and help as advisors. Professors such as Dr. Kong and Dr. Nichols were always there to lend a helping hand, and to tailor their classes to provide a substantial amount of material related to my research. Last but certainly not least, I would like to thank Hersh Patel for taking me under your wing and fueling a dormant fire that lay inside me for several years. If it wasn’t for you, then who knows where I would be today? I can’t forget about all the teachers throughout my life who ignited my infatuation to learn. Teachers such as Patricia McCoy, Meredith Griffith, Cary Garrett, Brad Goodson, Amanda Carmichael, Jenny Oliver, Julie Smith, Ellen Releford, Michael Stone, and Ron Nance. Thank you to every one of you for showing me kindness, love, and support.

Degree

Ph. D.; A dissertation submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Doctor of Philosophy.

Date

12-2024

Subject

Differential equations, Nonlinear--Numerical solutions; Gross-Pitaevskii equations; Reaction-diffusion equations

Keyword

Quantum Mechanics, Numerical Analysis, Reaction-Diffusion, Radial, Vortex, Spiral

Discipline

Applied Mathematics

Document Type

Doctoral dissertations

DCMI Type

Text

Extent

xv, 210 leaves

Language

English

Rights

http://rightsstatements.org/vocab/InC/1.0/

License

http://creativecommons.org/licenses/by-nc-nd/4.0/

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