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Selection of Marginally Stable Patterns by Boundary Conditions

  • Jacob Daniels
  • Simon Eskin
  • Tel Fahrendholz
  • Arnd Scheel
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Abstract

We study the effect of boundary conditions on the selection of wavenumbers in pattern-forming systems. Prior work established that, for a large class of boundary conditions, equilibria only within a narrow band of wavenumbers are compatible with the boundary. We demonstrate here that for a large class of initial conditions, solutions evolve towards an extremal member of this narrow band of equilibria. In fact, this extremal member is neutrally stable, nonlinearly unstable, and the convergence is realized through a series of snapping events at the boundary. We observe this behavior in simulations of the Swift-Hohenberg equation when the pattern forming region is confined to a large spatial interval through a parameter quench. We prove the convergence in a phase-diffusion approximation using comparison principles and astutely chosen subsolutions that mimic the snapping behavior.

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Cite

BibTeXRIS
BibTeX
@misc{daniels2026selection,
  title = {{Selection of Marginally Stable Patterns by Boundary Conditions}},
  author = {Jacob Daniels and Simon Eskin and Tel Fahrendholz and Arnd Scheel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15201},
  eprint = {2610.15201},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15201},
}
RIS
TY  - UNPB
TI  - Selection of Marginally Stable Patterns by Boundary Conditions
AU  - Jacob Daniels
AU  - Simon Eskin
AU  - Tel Fahrendholz
AU  - Arnd Scheel
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15201
UR  - https://arxiv.org/abs/2610.15201
AB  - We study the effect of boundary conditions on the selection of wavenumbers in pattern-forming systems. Prior work established that, for a large class of boundary conditions, equilibria only within a narrow band of wavenumbers are compatible with the boundary. We demonstrate here that for a large class of initial conditions, solutions evolve towards an extremal member of this narrow band of equilibria. In fact, this extremal member is neutrally stable, nonlinearly unstable, and the convergence is realized through a series of snapping events at the boundary. We observe this behavior in simulations of the Swift-Hohenberg equation when the pattern forming region is confined to a large spatial interval through a parameter quench. We prove the convergence in a phase-diffusion approximation using comparison principles and astutely chosen subsolutions that mimic the snapping behavior.
ER  -