Mathematics

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)