Contact Info

Primary Title: 
Postdoctoral Scholar
Email Contact: 
Office: 
KIDD 008
Office Hours: 

M 10am-11am, W 2pm-3pm, Th 3pm-4pm via Zoom

Bio: 

Patrik was an undergraduate at Oregon State University, and received his BS in Mathematics. Patrik received his PhD in Applied Mathematics from the University of Arizona working under the supervision of Ken McLaughlin and Vladimir Zakharov.

Education

PhD in Applied Mathematics, University of Arizona, 2018
MS in Applied Mathematics, University of Arizona, 2015
BS in Mathematics, Oregon State University, 2012

Research

Research Field: 
Applications of Complex Analysis to Mathematical Physics, Nonlinear Waves, Geometry, Probability, and Numerical Analysis.
Research Description: 

My research is on solitons, spinning things, donuts, and something random (sometimes on computers).

I use methods from complex analysis involving Riemann–Hilbert problems, nonlocal dbar problems, meromorphic functions on (large genus or infinite type) Riemann surfaces. I am also interested in solutions to partial differential equations based on this theory, including analytical methods and numerical methods based on nonlinear superposition principles.

One of my primary focus is on exact complex analytical solutions and numerical solutions to PDEs describing nonlinear waves. These include weakly nonlinear water waves, and weakly nonlinear electromagnetic waves.

My probability interests are on soliton gasses, Markov processes and integrable probability.

My geometry interests are on Riemann surfaces with holomorphic differential one forms, translation structures, and holomorphic line bundles on them.

I have taught classes on mathematical modeling, applied partial differential equations, complex variables, differential calculus, integral calculus, applied differential equations, matrix algebra and series/sequences. I am particularly interested teaching students how to apply abstract techniques learned in mathematics courses to important problems in science, engineering, and pure mathematics.

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

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

Unpublished Manuscripts:
Nabelek, P., Pickrell, D. ``Harmonic Maps and the Symplectic Category.'' (2014) (arXiv:1404.2899)

Research Group: