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 \cite{unbounded_domain_cadiot} to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and non-space group symmetries. We demonstrate our approach on the Swift-Hohenberg model. In particular, for a given symmetry group $\mathcal{G}$, we construct a natural Hilbert space $H_{\mathcal{G}}^l$ containing only functions with $\mathcal{G}$-symmetry. In this space, products and differential operators are well-defined allowing for the study of autonomous semi-linear PDEs. Depending on the properties of $\mathcal{G}$, we derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $u_0$. 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 $u_0$, the approximate inverse, and the computation of these bounds will depend on the properties of $\mathcal{G}$ and its maximal square lattice space subgroup, $\mathcal{H}$. More specifically, we consider three cases: $\mathcal{G}$ is a space group which can be represented on the square lattice, $\mathcal{G}$ is not a space group which can be represented on the square lattice and the symmetry of $\mathcal{H}$ isolates the solution, and where $\mathcal{G}$ is not a space group which can be represented on the square lattice and the symmetry of $\mathcal{H}$ does not isolate the solution. We demonstrate the methodology on the 2D 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 Github.