Solitons and Spinning Things

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)