2403.02949
Radial amplitude equations for fully localised planar patterns
Dan J. Hill, David J. B. Lloyd
correctmedium confidence
- Category
- math.DS
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper derives radial amplitude equations for stripes, hexagons/rhomboids, and a 12-fold quasipattern and provides explicit steady homoclinic solutions and Maxwell points. The candidate solution reproduces the same amplitude equations, existence conditions, Maxwell-point formula µ̂M = 8 ν̂^2/(9 a), and explicit homoclinic profiles (sech for cubic GL; a closed-form rational–sech^2 equivalent to the paper’s cosh form for quadratic–cubic). Boundary conditions, evenness, and reconstruction of the leading-order fully localized Swift–Hohenberg fields also agree. Minor differences are only in equivalent closed-form expressions and orientation choices; no substantive conflicts.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} strong field
\textbf{Justification:}
The manuscript introduces a clean and broadly applicable Bessel-function-based multiple-scales framework to derive radial amplitude equations for fully localized planar patterns and supplies explicit homoclinic pulses and Maxwell points. The analysis is sound and aligns with known results, and the extension to reaction–diffusion systems underscores generality. Minor clarifications (sign conventions, listing equivalent pulse formulas) would further improve readability.