Proving Symmetry of Localized Solutions and Application to Dihedral Patterns in the Planar Swift–Hohenberg PDE

Abstract. In this article, we extend the framework developed in to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and nonspace group symmetries. We demonstrate our approach on the Swift–Hohenberg model. In particular, for a given symmetry group [Formula: see text], we construct a natural Hilbert space [Formula: see text] containing only functions with [Formula: see text]-symmetry. In this space, products and differential operators are well-defined allowing for the study of autonomous semilinear PDEs. Depending on the properties of [Formula: see text], we derive a Newton–Kantorovich approach based on the construction of an approximate inverse around an approximate solution, [Formula: see text]. More specifically, combining a meticulous analysis and computer-assisted techniques, the Newton-Kantorovich approach is validated thanks to the computation of some explicit bounds. The strategy for constructing [Formula: see text], the approximate inverse, and the computation of these bounds will depend on the properties of [Formula: see text] and its maximal square lattice space subgroup, [Formula: see text]. More specifically, we consider three cases: [Formula: see text] is a space group which can be represented on the square lattice, [Formula: see text] is not a space group which can be represented on the square lattice and the symmetry of [Formula: see text] isolates the solution, and where [Formula: see text] is not a space group which can be represented on the square lattice and the symmetry of [Formula: see text] does not isolate the solution. We demonstrate the methodology on the 2 dimensional Swift–Hohenberg PDE by proving the existence of various dihedral localized patterns. The algorithmic details to perform the computer-assisted proofs can be found on Zenodo [D. Blanco and M. Cadiot, localizedpatternsshsymmetry.jl, 2025].