Hannes Uecker, Research

My research centers around the analysis of nonlinear PDE over unbounded domains, Mathematical Modeling and Numerical Simulation. Recently, pde2path [6,12, 9, 18, 17, 63] has been a focus. For an overview, my research can be arranged into 4 overlapping areas:

  1. Fluid flow, in particular film flow with a free boundary over inclined planes [56, 57, 53, 40, 42, 35, 46, 36, 32]
  2. Nonlinear Optics and related systems (NLS related problems) [60, 55, 48, 45, 44, 38, 34, 27]
  3. Dynamics of pattern formation ((dissipative) problems in which the Ginzburg–Landau equation and variants play important roles) [62, 59, 61, 51, 54, 41, 20, 66]
  4. Miscellaneous PDE systems from physics, chemistry and biology, e.g., Reaction–diffusion systems from surface catalysis, and plasmonic nano–particles, [58, 50, 52, 49, 39, 30, 31, 29, 18]

From the analytic point of view the focus are existence and stability problems and approximation by amplitude or modulation equations. Concerning stability the main issue are continuous spectra up to the imaginary axis, which for parabolic problems yields at most diffusive stability, characterized by polynomial decay rates of spatially algebraically localized perturbations. The amplitude equations are important for model reduction and unification.

Publications, last update 17. Juni 2019. Note: the linked preprints may differ slightly from the published versions.

Book: Nonlinear PDEs: A Dynamical Systems Approach, AMS, G raduate Studies in Mathematics 182, 2017, mit Guido Schneider. AMS

Preprints

  1. Snaking branches of planar BCC fronts in the 3D Brusselator, 2019, (mit D. Wetzel) preprint
  2. Defect–like structures and localized patterns in SH357, 2019, (accepted by PRE, mit E. Knobloch und D. Wetzel) preprint
  3. Steady bifurcations of higher multiplicity in pde2path, 2018, preprint
  4. Optimal fishery with Coastal Catch, 2018, (mit D. Grass und Th. Upmann) preprint

    Peer revieved journal publications and book chapters

    2016–...

  5. Continuation for thin film hydrodynamics and related scalar problems, in Comput. Methods Appl. Sci., 50, Springer, Cham, 2019 (mit S. Engelnkemper, S. V. Gurevich, D. Wetzel and U. Thiele), preprint.
  6. Hopf bifurcation and time periodic orbits with pde2path - algorithms and applications, Comm. in Comp. Phys., 2019, preprint, pde2path homepage.
    (This replaces the old version (2016), where the algorithms, applications, and implementations were reviewed in a single document. Now the paper focusses on the algorithms and applications, while the implementation is reviewed in a Tutorial on Hopf bifurcation; see also pde2path tutorials.
  7. Mean field approach to first and second order phase transitions in ionic liquids, PRE, 95:060201, 2017, preprint, (mit S. Bier, N. Gavish and A. Yochelis)
  8. Desertification by Front Propagation?, Journal of Theoretical Biology 418 (2017), 27–35, (mit Y. Zelnik, U. Feudel und E. Meron) preprint
  9. Optimal management and spatial patterns in a distributed shallow lake model, 1:1-21, 2017, (mit D. Grass), preprint
  10. Low regularity justification results for envelope approximations of nonlinear wave packets in periodic media, Asymptotic Analysis, 99 (1-2), pp. 53-65, 2016 (mit K. Matthies), preprint
  11. Sine-Gordon solitons in networks: Scattering and transmission at vertices, Europ. Phys. Letters 115: 50002, 2016 (mit Z. Sobirov, D. Babajanov, K. Nakamura and D. Matrasulov), preprint
  12. Optimal control and spatial patterns in a semi arid vegetation system, Natural Resource Modeling, 29:2, 229–258, 2016, preprint
  13. Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross-Pitaevskii Equation, Journal of Nonlinear Science, 26:3, 581–618, 2016, (mit T. Dohnal), preprint

    2011–2015

  14. Exact solutions of the Cauchy problem for the linearized KdV equation on metric star graphs, Uzbek Mathematical J, 2015:3, 143-154, (mit Z. Sobirov und M. Akhmedov), preprint
  15. Soliton transport in tubular networks: transmission at vertices in the shrinking limit, PRE 91, 023209, 2015, (mit D. Grieser, Z. Sobirov, D. Babajanov and D. Matrasulov), preprint
  16. Individual-based model for quorum sensing with background flow, Bull. Math. Biology, (mit J. Müller und B. Hense), 2014, preprint
  17. Numerical results for snaking of patterns over patterns in some 2D Selkov–Schnakenberg Reaction–Diffusion systems, SIADS, 13-1:94–128, 2014 (mit D. Wetzel) preprint,
  18. pde2path – a Matlab package for continuation and bifurcation in 2D elliptic systems, NMTMA (Numerical Mathematics: Theory, Methods and Applications), 7:58–106, 2014 (mit D. Wetzel und J. Rademacher) preprint, pde2path homepage
  19. Approximating the dynamics of communicating cells in a diffusive medium by ODEs -Â Homogenization with Localization, J. Math. Biology, 2012, (mit J. Müller) preprint
  20. Diffusive stability and diffusive mixing of periodic wave trains in reaction diffusion systems. JDE, 252, 3541–3574, 2012. (mit B. Sandstede, A. Scheel and G. Schneider) preprint
  21. Self-similar decay to the marginally stable ground state in a model for film flow over inclined wavy bottoms, EJDE, 2012:61, 1–51, 2012, (mit T. Häcker und G. Schneider) preprint
  22. Traveling interface modulations in the NH3+O2 reaction on a Rh(110) surface, Physical Chemistry Chemical Physics, 14, 5260–5264, 2012, (mit M. Rafti, F. Lovis, V. Krupennikova and R. Imbihl) preprint
  23. Corrigendum to 30, Proc. Royal Soc. Edinburgh, 141A, 819-824, 2011 (mit R. Kaiser), preprint
  24. Interfacing Fortran routines from Matlab in a simple and efficient way - with applications to ordinary and partial differential equations Summer School Modern Computational Science, Oldenburg 2011, Universitätsverlag Oldenburg, 2011, preprint,
  25. Local existence and uniqueness of solutions of the weak electrolyte model describing electro-convection in nematic liquid crystals. ZAMM, 91:3, 247-256, 2011 (mit W.-P. Düll und G. Schneider) preprint
  26. Erratum to (published version of) 34, Physica D, 240, 357-362, 2011 (mit T. Dohnal)preprint
  27. Interaction of modulated pulses in nonlinear oscillator chains, J. Diff. Eq. Appl., 17:3, 279-298, 2011 (mit G. Schneider und M. Wand) preprint

    2006-2010

  28. Amplitude equations – an invitation to multiple scale analysis Summer School Modern Computational Science, Oldenburg 2010, Universitätsverlag Oldenburg, 2010, preprint,
  29. Statistics for surface modes of nanoparticles with shape fluctuations, Commun. in Comp. Physics, 8, 1224, 2010, (mit F. Rüting)
  30. Well-posedness of some initial-boundary-value problems for dynamo-generated poloidal magnetic fields, Proc. Royal Soc. Edinburgh, 139A, 1209-1235, 2009 (mit R. Kaiser) preprint
  31. Perturbation theory for plasmonic eigenvalues, Phys. Rev. B, 80:245405, 2009, (mit D. Grieser, S. Biehs, O. Huth, F. Rüting, and M. Holthaus) preprint
  32. An integral boundary layer equation for film flow over inclined wavy bottoms, Phys. of Fluids, 21, 092105, 2009. (mit T. Häcker) preprint
  33. A short ad hoc introduction to spectral methods for parabolic PDE and the Navier–Stokes equations, in Summer School Modern Computational Science, Oldenburg 2009, Universitätsverlag Oldenburg, 2009 preprint, software
  34. Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevsky equation with a non-separable periodic potential, Physica D, 238: 860-879, 2009. (mit T. Dohnal) (updated version)
  35. Linear Resonance in viscous films on inclined planes. IJ Mutiphase Flow 34: 580–589, 2008. (mit A. Wierschem, V. Bontozoglou, C. Heining, und N. Aksel)
  36. A Hopf-bifurcation theorem for the vorticity formulation of the Navier-Stokes equations in 3, Comm. PDE 33:5, 772–783, 2008. (mit A. Melcher and G. Schneider) preprint
  37. A remark about the justification of the nonlinear SchrÃdinger equation in quadratic spatially periodic media, ZAMP, 59:1-4, 2008 (mit C. Blank, M. Chirilus-Bruckner, C. Chong, V. Lescarret and G. Schneider) preprint
  38. Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves, Math. Anal. Appl. 347, 304Â314 (2008), (mit M. Chirilus-Bruckner, C. Chong, and G. Schneider) preprint
  39. Mass transport of alkali metal with pulses in a surface reaction, Phys. Rev. E, 78:055203(R), 2008, (mit M. Hinz, R. Imbihl, Y. Keverekidis und Q. Liang) preprint
  40. Long time persistence of KdV solitons as transient dynamics in a model of inclined film flow, Proc. Roy. Soc. Edinb., 137A: 133–146, 2007. (mit R.L. Pego and G. Schneider) preprint
  41. Exponential averaging and traveling waves in rapidly varying periodic media, Mathematische Nachrichten, 280:4, 408-422, 2007 (mit K. Matthies and G. Schneider) preprint
  42. Self–similar decay of localized perturbations of the Nusselt solution for the Navier–Stokes equations on an inclined plane, Arch.Rat.Mech.Anal. 184:3, 401–447, 2007 preprint
  43. The amplitude equations for the first instability of electro-convection in nematic liquid crystals in case of two unbounded space directions, Nonlinearity 20:1361-1386, 2007. (mit G. Schneider) preprint
  44. On the interaction of modulating pulses with different carrier waves, Mathematical Methods Applied Sciences, 30:15, 1965–1978, 2007. (mit M. Chirilus-Bruckner und G. Schneider) preprint
  45. The mathematics of light pulses in dispersive media, Jahresberichte der DMV, 109:3, 139-161, 2007 (mit G. Schneider) preprint
  46. A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom. EJDE 118, 1-18, 2007, (mit A. Wierschem) preprint
  47. Simulation of the life cycle of adsorbate islands on the Pt(100) surface during the NO+NH3 reaction. Chem. Phys. Letters, 421:4–6, 577–583. 2006. (mit M. Rafti, J. L. Vicente and R. Imbihl) preprint
  48. Justification of the Nonlinear Schrödinger equation in spatially periodic media, ZAMP, 57:6, 905–939, 2006 (mit K. Busch, G. Schneider and L. Tkeshelashvili) preprint

    2001-2005

  49. Pattern formation for NO+NH3 on Pt(100) – 2-dimensional numerical results, Phys. Rev.E, 71, 016207, 2005. preprint
  50. Standing waves, clustering, and phase waves in 1d-simulations of kinetic relaxation oscillations in NO+NH3 on Pt(100) coupled by diffusion. Physica D 190:249–265, 2004. preprint
  51. Stable transport of information near essentially unstable localized structures. Discrete Contin.Dyn.Syst.B 4:2, 349–390, 2004. (mit Th. Gallay und G. Schneider) preprint
  52. Relaxed plasma vacuum states in cylinders. Q.J. Mech. Appl. Math., 57:1, 1–17, 2004. (mit R. Kaiser) preprint
  53. Self–similar decay of localized perturbations in the Integral Boundary Layer equation. Journal Diff. Eq., 207:2, 407–422, 2004.preprint
  54. Validity of the Ginzburg-Landau approximation in pattern forming systems with time periodic forcing. Dynamics and Bifurcations of Patterns in Dissipative Systems, World Scientific, 39–57, 2004. (mit N. Breindl und G. Schneider) preprint
  55. Existence and stability of modulating pulse solutions in Maxwell’s equations describing nonlinear optics. ZAMP 54, 677-712, 2003. (mit G. Schneider) preprint
  56. Approximation of the Integral Boundary Layer equation by the Kuramoto–Sivashinsky equation. SIAM J. Appl. Math. 63:4, 1359–1377, 2003. preprint
  57. Almost global existence and transient self similar decay for Poiseuille flow at criticality over exponentially long times. Physica D 185:3–4, 209–226, 2003. (mit G. Schneider) preprint
  58. Adiabatic reduction and hysteresis of the LFI–model of NO+NH3 on Pt(100). Chem. Phys. Let., 382, 232–244, 2003. (mit R. Imbihl, M. Rafti, I.M. Iruzun, J.L. Vicente und E.E. Mola) preprint
  59. Stability and diffusive dynamics on unbounded domains. In Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems, ed. B. Fiedler, Springer, 563-584, 2001. (mit A. Mielke und G. Schneider)preprint
  60. Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems. Asymptot. Anal. 28:2, 163-180, 2001. (mit G. Schneider) preprint
  61. Stable modulating multi-pulse solutions for dissipative systems with resonant spatially periodic forcing. J. Nonlin. Sci.,11:2, 89-121, 2001. preprint
  62. Diffusive stability of rolls in the two-dimensional real and complex Swift–Hohenberg equation. Comm. PDE 24:11&12, 2109-2146, 1999.preprint

    Proceedings

  63. pde2path – V2 : multi-parameter continuation and periodic domains, 2014, in H. Ecker, A. Steindl, S. Jakubek, eds, ENOC 2014 - Proceedings of 8th European Nonlinear Dynamics Conference, ISBN: 978-3-200-03433-4. (mit T. Dohnal, J. Rademacher und D. Wetzel), preprint, pde2path homepage
  64. Modeling of film flows over inclined wavy bottoms, PAMM 8:10721-10722, 2009 (mit T. Häcker)
  65. Resonance in viscous film flow over topography PAMM 7:4100025–4100026, 2008 (mit C. Heining, A. Wierschem, V. Bontozoglou und N. Aksel)
  66. Mathematical theory for the Ginzburg-Landau approximation in pattern forming systems with time-periodic forcing – with applications to electro-convection in nematic liquid crystals. Proceedings of Equaiff–11: 507–517, 2005. (mit N. Breindl and G. Schneider) preprint
  67. Local in time nonlinear stability of pulses in an unstable medium. XIVth International Congress on Mathematical Physics, Lisboa 2003, 296–303, World Scientific, 2005. (mit R.L. Pego und G. Schneider)
  68. Rolls and modulating pulses in Swift–Hohenberg type of equations, in International Conference on Differential Equations, Equadiff 99, World Scientific, 408-413, 2000.

Thesis

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