Author: nabelekp

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 nabelekp@oregonstate.edu.

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.

Webpages I Like: ONR: Atlas of Internal Waves, SIAM, AMS, NOAA (Ocean and Coasts).

Publications:

7) Nabelek, P.V. “On solutions to the nonlocal dbar-problem and (2+1) dimensional completely integrable systems.” Lett Math Phys 111, 16 (2021). https://doi.org/10.1007/s11005-021-01353-w

6) Nabelek, P.V. “Algebro-geometric finite gap solutions to the Korteweg–de Vries equation as primitive solutions.” Phys D 414, 132709 (2020). https://doi.org/10.1016/j.physd.2020.132709

5) Nabelek, P.V., Zakharov, V.E. “Solutions to the Kaup–Broer system and its (2＋1) dimensional integrable generalization via the dressing method.” Phys D 409, 132478 (2020). https://doi.org/10.1016/j.physd.2020.132478. 

4) Dyachenko, S.A., Nabelek, P., Zakharov, D.V, Zakharov, V.E. “Primitive solutions of the Korteweg–de Vries equation.” Theor Math Phys 202, 334–343 (2020). https://doi.org/10.1134/S0040577920030058

3) McLaughlin, K.T-R, Nabelek, P.V. “A Riemann–Hilbert Problem Approach to Infinite Gap Hill’s Operators and the Korteweg–de Vries Equation.” Int Math Res Not 2, 1288–1352 (online 2019, print 2021). https://doi.org/10.1093/imrn/rnz156.

2) Nabelek, P., Zakharov, D., Zakharov, V. “On symmetric primitive potentials.” J Int Sys, 4:1, xyz006 (2019). https://doi.org/10.1093/integr/xyz006

1) Dissertation: Applications of Complex Variables to Spectral Theory and Completely Integrable Partial Differential Equations. https://repository.arizona.edu/handle/10150/627724

Unpublished Manuscripts:

Nabelek, P., Pickrell, D. “Harmonic Maps and the Symplectic Category.” (2014) (arXiv:1404.2899)

Nabelek, P. “Distributions Supported on Fractal Sets and Solutions to the Kadomtsev–Petviashvili Equation.” (2020) (arXiv:2009.05864)