I am interested in the existence, stability, and dynamics of nonlinear waves in partial differential equations arising from applications.
1. Subharmonic Dynamics of Wave Trains in Systems of Viscous Conservation Laws.
With Mathew Johnson.
In preparation.
1. On the Modulation of Wave Trains in the Ostrovsky Equation.
With Mathew Johnson and Jeffrey Oregero.
Submitted.
(arXiv)
8. Convergence of Linear Solutions Through Convergence of Periodic Initial Data.
With Harrison Gaebler.
Journal of Mathematical Analysis and Applications,
Vol. 552 No. 1 Article 129698: 17pp, 2025.
(doi,
arXiv)
7. Nonlinear Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves.
With Mariana Haragus, Mathew Johnson, and Björn de Rijk.
Communications in Mathematical Physics,
Vol. 405 Issue 10 Article 227: 32pp, 2024.
(doi,
arXiv)
6. Nonlinear Modulational Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves.
With Mariana Haragus, Mathew Johnson, and Björn de Rijk.
Annales de l'Institut Henri Poincaré, Analyse Non Linéaire,
Vol. 40 No. 4: pp. 769-802, 2023.
(doi,
arXiv)
5. Rigorous Justification of the Whitham Modulation Equations for Equations of Whitham-Type.
With William Clarke and Robert Marangell.
Studies in Applied Mathematics,
Vol. 149 Issue 2: pp 297-323, 2022.
(doi,
arXiv)
4. Subharmonic Dynamics of Wave Trains of the Korteweg-de Vries/Kuramoto-Sivashinsky Equation.
With Mathew Johnson.
Studies in Applied Mathematics,
Vol. 148 Issue 3: pp 1274-1302, 2022.
(doi,
arXiv)
3. Subharmonic Dynamics of Wave Trains in Reaction Diffusion Systems.
With Mathew Johnson.
Physica D,
Vol. 422 Article 132891: 24pp, 2021.
(doi,
arXiv)
2. Linear Modulational and Subharmonic Dynamics of Spectrally Stable Lugiato-Lefever Periodic Waves.
With Mariana Haragus
and Mathew Johnson.
Journal of Differential Equations,
Vol. 280: pp 315-354, 2021.
(doi,
arXiv)
1. Modulational Instability of Viscous Fluid Conduit Periodic Waves.
With Mathew Johnson.
SIAM Journal on Mathematical Analysis,
Vol. 52 Issue 1: pp 277-305, 2020.
(doi,
arXiv)
Modulational and Subharmonic Dynamics of Periodic Waves.
ProQuest,
KU ScholarWorks
206pp, 2021.
(URI)
Circles in \(\mathbb{F}_q^2\).
With Jacob Haddock, John Pope, and Jeremy Chapman.
Aletheia,
Vol. 2 Issue 1: 7pp, 2017.
(PDF, alternative proof of Lemma 2.6.)
In this paper, we investigate how circles may be determined in \(\mathbb{F}_q^2\), the two-dimensional vector space over the finite field \(\mathbb{F}_q\). This investigation is complicated by the fact that two distinct points may have zero distance between them. Nonetheless, we demonstrate that a circle of nonzero radius may be uniquely determined by three distinct, noncollinear points, which pairwise have nonzero distance between them.