Pattern-Forming Partial Differential Equations and Optimal Point Configurations

Abstract: This thesis asks whether pattern-forming partial differential equations can serve as dynamical routes to optimal point configurations on the sphere. At the planar level, the Lifshitz–Petrich equation is shown to self-organize into near-optimal hexagonal packings from random initial conditions. At the spherical level, steady states of the Swift–Hohenberg equation are found to correspond closely to Thomson-problem optima across a range of particle numbers, suggesting that both systems converge to a shared class of quasi-uniform hexagonal configurations — each necessarily carrying exactly twelve pentagonal disclinations.