From approximate dihedral patches to exact PDE solutions

Extend the Galerkin and radial‑normal‑form constructions of localized dihedral‑symmetric patches near onset to prove the existence of fully localized dihedral patterns as exact steady solutions of the planar Swift–Hohenberg equation and of general reaction–diffusion systems.

Background

A recent approximate theory constructs numerous families of localized dihedral patches via finite Fourier–cosine Galerkin truncations and radial reductions near the Turing point.

Bridging these approximate constructions to rigorous existence results for the full PDE remains unresolved.