This paper derives radial ring formation as a natural consequence of pattern-forming dynamics applied to a ΔΦ-based scalar field. Using a Swift–Hohenberg framework, the analysis shows that above a critical instability threshold, the system selects a dominant wavelength determined by the most unstable wavenumber. In polar geometry, this produces concentric ring structures as axisymmetric eigenmodes. Analytical predictions of wavelength are numerically verified with high precision. The result establishes a conditional structural principle for radial pattern formation in constrained fields, with limitations and assumptions explicitly stated.
Thomas S. Mitchell (Tue,) studied this question.