The UTC Graduate School is pleased to announce that James Cummins will present Doctoral research titled, Radial, Vortex, and Spiral Solutions to the Nonlinear Schrödinger Equation and Other Reaction--Diffusion Systems on 10/03/2024 at 4:30 in Lupton 389. Everyone is invited to attend. 

Computational Science

Chair: Christoper Cox

Co-Chair: Boris Belinskiy

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 phase dependence), and spiral solutions (which depend on phase). The study considers both 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 constant phase. Furthermore, we show that the spiral solutions to the nonlinear Schrödinger equation amount to solving a system of nonlinear ordinary differential equations. This system is shown to be a special case of the $\lambda$--$\omega$ reaction--diffusion system.