My name is Patrik Nabelek, and this is my personal webpage. My research is on mathematical physics generally, and on nonlinear differential equations describing wave motion with a focus on the formation and replication of hydrodynamic soliton gasses in shallow water. I am also interested in nonlinear electromagnetic waves and soliton gas formation within fiber optic cables. I am interested in completely integrable Hamiltonian systems, translation surfaces, and donuts. I am also interested in probability and statistical models in classical linear and nonlinear wave mechanics. If you would like to contact me, please send me an email at email@example.com.
In my research, I use methods from functional analysis, complex analysis, spectral theory, and geometry. I am also interested in the use finite difference methods and psuedo-spectral methods to solve problems relevant in ocean engineering and nonlinear optics. One of my focuses is on completely integrable systems that describe phenomena in classical, quantum, and wave mechanics.
I am also interested in real and complex manifolds and surfaces from the points of view of both algebraic and Riemannian geometry. I focus on Riemann surfaces, and their complex structures, translation structures, flat cone metrics metrics, and their harmonic maps into semi-simple Lie groups. But I am also interested in holomorphic and meromorphic functions on two dimensional complex manifolds. I study these topics in connection with the theory of completely integrable Hamiltonian systems in finite and infinite dimensions, and on the study of soliton gasses.
I am particularly interested in: the Korteweg–de Vries equation, the one dimensional Boussinesq equation, and Kaup–Broer system for shallow water; the cubic nonlinear Schrodinger equation for deep water waves; the Burgers equation for gas dynamics. I am also interested in the Hamiltonian formulation of the deep and shallow water wave wave systems, and the complete integrability conjecture for one dimensional deep water wave system.
I currently work as a postdoc at Oregon State University teaching classes on these subjects senior level mathematics undergraduate students and masters level engineering graduate students. I received a PhD in Applied Mathematics from UA in 2018 working under Vladimir Zakharov and Ken McLaughlin, and a BS in Mathematics from Oregon State University.