Hexagonal spike clusters for some PDE's in 2D
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Bibliographic record
Abstract
<p style='text-indent:20px;'>We study hexagonal spike cluster patterns for Gierer-Meinhardt reaction-diffusion system with a precursor on all of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^2 $\end{document}</tex-math></inline-formula>. These clusters consist of <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> spikes which form a nearly hexagonal lattice of a finite size. The lattice density is locally nearly constant, but globally non-uniform. We also characterize a similar hexagonal spike cluster steady state for a simple elliptic PDE <inline-formula><tex-math id="M3">\begin{document}$ 0 = \Delta u - u +u^2 + \varepsilon |x|^2 $\end{document}</tex-math></inline-formula> with a small "confinement well" <inline-formula><tex-math id="M4">\begin{document}$ \varepsilon |x|^2 $\end{document}</tex-math></inline-formula>. The key idea is to explicitly exploit the local hexagonality structure to asymptotically approximate the solution using certain lattice sums. In the limit of many spikes, we derive the effective spike density as well as the cluster radius. This effective density is a solution to a certain separable first-order ODE coupled to an integral boundary condition.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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