Publications // University of Oldenburg

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A. Meiners and H. Uecker, "Helfrich Cylinders—Instabilities, Bifurcations, and Amplitude Equations" SIAM Journal on Applied Dynamical Systems, vol. 25, iss. 1. 2026.
doi: 10.1137/25M173973X
@article{hcyl, author = {Meiners, Alexander and Uecker, Hannes}, title = {Helfrich Cylinders—Instabilities, Bifurcations, and Amplitude Equations}, journal = {SIAM Journal on Applied Dynamical Systems}, volume = {25}, number = {1}, pages = {439-484}, year = {2026}, doi = {10.1137/25M173973X}, URL = { https://cloud.uol.de/s/ZTXtZoJiaJBKbCH?dir=/&editing=false&openfile=true}, eprint = { https://epubs.siam.org/doi/pdf/10.1137/25M173973X}, abstract = { Abstract.Combining local bifurcation analysis with numerical continuation and bifurcation methods we study bifurcations from cylindrical vesicles described by the Helfrich equation with volume and area constraints, with a prescribed periodicity along the cylindrical axis. The bifurcating solutions are in two main classes, axisymmetric (pearling), and nonaxisymmetric (coiling, buckling, and wrinkling), and depending on the spontaneous curvature and the prescribed periodicity along the cylinder axis we obtain different stabilities of the bifurcating branches, and different secondary bifurcations. } }
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F. Al Saadi, E. Knobloch, A. Meiners, and H. Uecker, "Breathers and mixed oscillatory states near a Turing-Hopf instability in a two-component reaction-diffusion system" Physica D: Nonlinear Phenomena, vol. 472, p. 134482. 2025.
doi: 10.1016/j.physd.2024.134482
@article{ALSAADI2025134482, title = {Breathers and mixed oscillatory states near a Turing-Hopf instability in a two-component reaction-diffusion system}, journal = {Physica D: Nonlinear Phenomena}, volume = {472}, pages = {134482}, year = {2025}, issn = {0167-2789}, doi = {https://doi.org/10.1016/j.physd.2024.134482}, url = {https://cloud.uol.de/s/t2fDjtmjrwze2ra}, author = {Fahad {Al Saadi} and Edgar Knobloch and Alexander Meiners and Hannes Uecker}, keywords = {Localized oscillatory states, Breathing, Mixed modes, Numerical bifurcation and continuation, Snaking, Isolas, Zipping-up}, abstract = {Numerical continuation is used to study the interaction between a finite wave number Turing instability and a zero wave number Hopf instability in a two-species reaction-diffusion model of a semiconductor device. The model admits two such codimension-two interactions, both with a subcritical Turing branch that is responsible for the presence of spatially localized Turing states. The Hopf branch may also be subcritical. We uncover a large variety of spatially extended and spatially localized states in the vicinity of these points and by varying a third parameter show how disconnected branches of time-periodic spatially localized states can be â€œzipped upâ€ into snaking branches of time-periodic oscillations. These are of two types: a Turing state embedded in an oscillating background, and a breathing Turing state embedded in a non-oscillating background. Stable two-frequency states resembling a mixture of these two states are also identified. Our results are complemented by direct numerical simulations. The findings explain the origin of the large multiplicity of localized steady and oscillatory patterns arising from the Turingâ€“Hopf interaction and shed light on the competition between them.} }
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A. Meiners and H. Uecker, "Numerical Continuation and Bifurcation for Differential Geometric PDEs" Numerical Mathematics: Theory, Methods and Applications, vol. 17, iss. 3. 2024.
doi: 10.4208/nmtma.OA-2024-0005
@article{NMTMA-17-3, author = {Meiners, Alexander and Uecker, Hannes}, title = {Numerical Continuation and Bifurcation for Differential Geometric PDEs}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {3}, pages = {555--606}, abstract = {We describe some differential geometric bifurcation problems and their treatment in the Matlab continuation and bifurcation toolbox pde2path. The continuation steps consist in solving the PDEs for the normal displacement of an immersed surface $X âŠ‚\mathbb{R}^3,$ with bifurcation detection and possible subsequent branch switching. The examples include minimal surfaces such as Enneperâ€™s surface and a Schwarz-P-family, some non-zero constant mean curvature surfaces such as liquid bridges, and some 4th order biomembrane models. In all of these we find interesting symmetry-breaking bifurcations. A few of these are (semi)analytically known and hence used as benchmarks. }, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2024-0005}, url = {https://cloud.uol.de/s/FBYieM3nAFZp6CQ} }
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A. Meiners and B. Vertman, "Spectral zeta function on discrete tori and Epstein-Riemann Hypothesis" Journal of Number Theory, vol. 244. 2023.
doi: 10.1016/j.jnt.2022.08.004
@article{MEINERS2023418, title = {Spectral zeta function on discrete tori and Epstein-Riemann Hypothesis}, journal = {Journal of Number Theory}, volume = {244}, pages = {418-461}, year = {2023}, issn = {0022-314X}, doi = {https://doi.org/10.1016/j.jnt.2022.08.004}, url = {https://www.sciencedirect.com/science/article/pii/S0022314X2200186X}, author = {Alexander Meiners and Boris Vertman}, keywords = {Epstein zeta function, Spectral zeta function, Epstein-Riemann Hypothesis},

Tutorials

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A. Meiners and H. Uecker, Differential geometric bifurcation problems in pde2path -- algorithms and tutorial examples.
@misc{geomtut, author = {Meiners, A. and Uecker, H.}, TITLE = {Differential geometric bifurcation problems in pde2path -- algorithms and tutorial examples}, url= "https://cloud.uol.de/s/iMDai3idbz3TpPx", YEAR = {2023} }

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