Tomas Dohnal

Tomáš Dohnal
Angewandte Analysis
Institut für Mathematik
MLU Halle-Wittenberg
Institut für Mathematik Halle Universitaet Halle

Contact:Why do math?

Office hours: Dienstags 11:00-12:00 und nach Vereinbarung (im Büro)                      

Mailing Address:

Tomáš Dohnal
Institut für Mathematik
Martin-Luther-Universität Halle-Wittenberg
06099 Halle (Saale), Germany

Current and recent teaching:

TD 
approximate solitary wave in the 2D periodic NLS

Research interests:      broadly speaking "PDEs"

  • waves, dispersive PDEs
    • rigorous asymptotics of wavepackets in nonlinear problems
    • solitary waves in periodic structures and at surfaces
    • spectral problems
  • PDE bifurcation problems
    • bifurcation of nonlinear solutions from spectrum
    • bifurcation in PT-symmetric (non-selfadjoint) problems
  • numerics
    • simulation of nonlinear waves 
    • bifurcation package PDE2PATH for elliptic PDEs
Projects/Grants:

Collaborators:
PhD Students:
Bachelor / Master students: link

Publications:

Preprints:

  1. T. Dohnal and Runan He, ``Bifurcation and Asymptotics of Cubically Nonlinear Transverse Magnetic Surface Plasmon Polaritons,'' accepted to J. Math. Anal. Appl., 2024. (arXiv:2311.17838)
  2. M. Brown, T. Dohnal, M. Plum, and I. Wood, ``Spectrum of the Maxwell Equations for a Flat Interface between Homogeneous Dispersive Media," submitted, 2022. (arXiv:2206.02037)

Journal articles:

  1. T. Dohnal, D. Pelinovsky, and G. Schneider, ``Traveling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media," Nonlinearity, 37, 055005 (2024). (https://doi.org/10.1088/1361-6544/ad3097, arXiv:2304.06214)
  2. T. Dohnal, M. Ionescu-Tira, and M. Waurick, ``Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface,"  J. Diff. Eq., 383 (25), 24-77 (2023). (https://doi.org/10.1016/j.jde.2023.11.005)(arXiv:2301.10099)
  3. T. Dohnal, R. Schnaubelt, and D.P. Tietz, ``Rigorous Envelope Approximation for Interface Wave-Packets in Maxwell's Equations with 2D Localization,'' SIAM J. Math. Anal., 55 (6), 6898-6939 (2023).  (https://epubs.siam.org/doi/10.1137/22M1501611) (arXiv:2206.03154)
  4. T. Dohnal, G. Romani, and D.P. Tietz, ``A quasilinear transmission problem with application to Maxwell equations with a divergence-free D-field," J. Math. Anal. Appl., 511 (1), 126067 (2022). (https://doi.org/10.1016/j.jmaa.2022.126067) (arXiv:2109.08513)
  5. T. Dohnal, G. Romani, ``Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations," Nonlinearity, 34 (8),  5261-5318 (2021). (https://iopscience.iop.org/article/10.1088/1361-6544/ac0485, arXiv:2010.03473)
  6. T. Dohnal and L. Wahlers, ``Bifurcation of Gap Solitons in Coupled Mode Equations in d Dimensions,'' J Dyn Diff Equat, 2021. (https://doi.org/10.1007/s10884-021-09971-7, arXiv:1903.02631)
  7. T. Dohnal and G. Romani, ``Eigenvalue Bifurcation in Doubly Nonlinear Problems with an Application to Surface Plasmon Polaritons,'' Nonlinear Differ. Equ. Appl. 28 (9) (2021). (https://doi.org/10.1007/s00030-020-00668-2, arxiv:2002.08674v4) Note: The arXiv version is a revised and corrected one. It includes the corrections from the erratum:
    • ``Correction to: Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons,'' Nonlinear Differ. Equ. Appl. 30, 9 (2023). https://doi.org/10.1007/s00030-022-00815-x.
  8. T. Dohnal and L. Wahlers, ``Coupled Mode Equations and Gap Solitons in Higher Dimensions,''  J. Diff. Eq., 269 (3), 2386–2418 (2020). (https://doi.org/10.1016/j.jde.2020.01.037, arXiv:1810.04944)
  9. A. Mannan, S. Sultana, R. Schlickeiser, and T. Dohnal, ``Three-dimensional self-gravito-acoustic solitary waves in a degenerate quantum plasma system'', Plasma Phys. Rep., 46 (2),195–199 (2020). (https://doi.org/10.1134/S1063780X20020075)
  10. A. Mannan and T. Dohnal, ``(3+1)-dimensional cylindrical Korteweg-de Vries equation in a self-gravitating degenerate quantum plasma system,'' Physics of Plasmas 27, 012102 (2020). (doi.org/10.1063/1.5129799)
  11. T. Dohnal and D. Pelinovsky, ``Bifurcation of nonlinear bound states in the periodic Gross-Pitaevskii equation with PT-symmetry,''  Proc. R. Soc. Edinb. A, 150 (1), 171-204 (2020). (doi.org/10.1017/prm.2018.83, arXiv:1702.0346)
  12. T. Dohnal and D. Rudolf, ``NLS approximation for wavepackets in periodic cubically nonlinear wave problems in Rd,'' Applicable Analysis, 99 (10), 1685-1723 (2020). (doi.org/10.1080/00036811.2018.1544620, arXiv:1710.07077)
  13. T. Dohnal and B. Schweizer, ``A Bloch wave numerical scheme for scattering problems in periodic wave-guides,'' SIAM J. Num. Anal., 56 (3), 1848–1870 (2018). (https://doi.org/10.1137/17M1141643, arXiv:1708.06427)
  14. T. Dohnal and L. Helfmeier, ``Justification of the Coupled Mode Asymptotics for Localized Wavepackets in the Periodic Nonlinear Schrödinger Equation,'' J. Math. Anal. Appl. 450, 691-726 (2017). (https://doi.org/10.1016/j.jmaa.2017.01.039, arxiv:1602.04121)
  15. T. Dohnal and P. Siegl, ``Bifurcation of eigenvalues in nonlinear problems with antilinear symmetry,''  J. Math. Phys 57, 093502 (2016). (https://doi.org/10.1063/1.4962417, arXiv:1504.00054)
  16. T. Bartsch, T. Dohnal, M. Plum, and W. Reichel, ``Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media,'' Nonlinear Differ. Equ. Appl. (2016) 23: 52. (https://doi.org/10.1007/s00030-016-0403-0, arXiv:1411.7153)
  17. T. Dohnal and H. Uecker, ``Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation,''  J. Nonlin. Sci. 26(3):581-618 (2016). (https://doi.org/10.1007/s00332-015-9281-6, arXiv: 1409.4199)
  18. T. Dohnal, A. Lamacz, and B. Schweizer, ``Dispersive homogenized models and coefficient formulas for waves in general periodic media,'' Asymptotic Analysis 93, 21-49 (2015). (arXiv:1401.7839)
  19. T. Dohnal, A. Lamacz, and B. Schweizer, ``Bloch-wave homogenization on large time scales and dispersive effective wave equations,'' Multiscale Model. Simul. 12, 488-513 (2014). (arXiv:1302.4865)
  20. T. Dohnal, ``Traveling Solitary Waves in the Periodic Nonlinear Schrödinger Equation with Finite Band Potentials,'' SIAM Appl. Math. 74, 306-321 (2014). (arXiv:1305.3504)
  21. T. Dohnal, K. Nagatou, M. Plum and W. Reichel, ``Interfaces Supporting Surface Gap Soliton Ground States in the 1D Nonlinear Schrödinger Equation,'' J. Math. Anal. Appl. 407 , 425-435 (2013). (arXiv:1202.3588)
  22. T. Dohnal and W. Dörfler, ``Coupled Mode Equation Modeling for Out-of-Plane Gap Solitons in 2D Photonic Crystals,'' Multiscale Model. Simul. 11, 162-191 (2013). (arXiv:1202.3583)
  23. T. Dohnal and D. Pelinovsky, ``Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential,'' Phys. Rev. E 85:026605 (2012). (arXiv:1110.3780)
  24. T. Dohnal, M. Plum and W. Reichel, ``Surface gap soliton ground states for the nonlinear Schrödinger equation,'' Comm. Math. Phys. 308, 511-542 (2011). (https://doi.org/10.1007/s00220-011-1320-z, arXiv:1011.2886)
  25. E. Blank and T. Dohnal, "Families of Surface Gap Solitons and their Stability via the Numerical Evans Function Method," SIAM J. Appl. Dyn. Syst. 10, 667-706 (2011). (https://doi.org/10.1137/090775324, arXiv:0910.4858)
  26. T. Dohnal and H. Uecker, ``Erratum to `Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential' by T. Dohnal and H. Uecker [Physica D 238 (2009), 860-879],'' Physica D 240, 357-362 (2011).
  27. T. Dohnal, M. Plum and W. Reichel, ``Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation,'' SIAM J. Math. Anal. 41, 1967-1993 (2009). (arXiv:0811.4514)
  28. T. Dohnal, ``Perfectly Matched Layers for Coupled Nonlinear Schrödinger Equations with Mixed Derivatives,'' J. Comput. Phys. 228, 8752–8765 (2009). (arXiv:0905.2321)
  29. A. Peleg, Y. Chung, T. Dohnal, and Q. M. Nguyen, ``Diverging probability density functions for flat-top solitary waves,'' Phys. Rev. E 80:026602 (2009). (arXiv:0906.3001)
  30. T. Dohnal and H. Uecker, ``Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential,'' Physica D 238, 860-879 (2009). (arXiv:0810.4499) Note: The arXiv version is a largely revised and corrected one.
  31. T. Dohnal, D. Pelinovsky and G. Schneider, ``Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential,'' J. Nonlin. Sci. 19, 95-131 (2009). (arXiv:0707.3731)
  32. T. Dohnal and D. Pelinovsky, ``Surface Gap Solitons at a Nonlinearity Interface," SIAM J. Appl. Dyn. Syst. 7, 249-264 (2008). (arXiv:0704.1742)
  33. T. Dohnal and T. Hagstrom, ``Perfectly matched layers in photonics computations: 1D and 2D Nonlinear Coupled Mode Equations," J. Comput. Phys. 223, 690-710 (2007).
  34. A.B. Aceves and T. Dohnal, ``Finite dimensional model for defect-trapped light in planar periodic nonlinear stuctures," Opt. Lett. 31, 3013-3015 (2006).
  35. A. Peleg, T. Dohnal, and Y. Chung, ``Effects of dissipative disorder on front formation in pattern forming systems,'' Phys. Rev. E 72:027203 (2005).
  36. T. Dohnal and A.B. Aceves, ``Optical soliton bullets in (2+1)D nonlinear Bragg resonant periodic geometries,'' J. Yang, editor, Nonlinear Wave Phenomena in Periodic Photonic Structures, Studies in Applied Math. 115:209-232 (2005).

Conference proceedings:

  1. T. Dohnal, J. Rademacher, H. Uecker, D. Wetzel, pde2path 2.0: multi-parameter continuation and periodic domains, in H. Ecker, A. Steindl, S. Jakubek, eds, ENOC 2014 - Proceedings of 8th European Nonlinear Dynamics Conference, ISBN: 978-3-200-03433-4.
  2. A.B. Aceves and T. Dohnal, ``Stopping and bending light in 2D photonic structures,'' Proceedings of OSA topical meeting on Nonlinear Guided Waves and their Applications, Toronto, March 2004.
  3. A.B. Aceves and T. Dohnal, ``Stopping and bending light in 2D photonic structures,'' in `` Nonlinear Waves: Classical and Quantum Effects,'' p. 293 - 302, F. Kh. Abdullaev and V.V. Konotop (eds.), Kluwer, 2004.

Dissertation:  Optical bullets in (2+1)D photonic structures and their interaction with localized defects, PhD dissertation, Univ. of New Mexico, 2005.

Habilitation:   Localized Waves in Periodic Structures, Karslruhe Institute of Technology, May 2012.

Software Package:

  • PDE2PATH, a Matlab package for continuation and bifurcation in 2D elliptic systems, with J. Rademacher, H. Uecker, and D. Wetzel. (manual also on arXiv)

Past teaching:

Short (professional) history:
"WAS IST ...?" - talks for a broad math-audience

ICIAM 07 Minisymposia: 
Strongly Nonlinear Phenomena in Optics, BECs, Hydrodynamics and Biology

Curriculum vitae: pdf file

 

Past affiliations:
TU Dortmund
 Department of Mathematics, Technical University in Dortmund, Germany


Department of Mathematics, Karlsruhe Institute of Technology, Germany


Alexander von Humboldt Foundation, Germany
ETH Zuerich

Seminar for Applied Mathematics, ETH Zurich, Switzerland

UNM logo

Department of Mathematics and Statistics, University of New Mexico, USA

LANL logo

Los Alamos National Laboratory (Group T7), Los Alamos, New Mexico, USA

TUL logo

Technical University in Liberec, Czech Republic

Nonprofessional history:

I am married and have three kids. I come from Jablonec nad Nisou, a town in the north of the Czech Republic. I was, however, born in Prostejovicky,  a  beautiful village in Moravia - the eastern part of the country. The tiny village with about 280  people is close to a town called Prostejov .