Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
Dominic Blanco, Matthieu Cadiot
TL;DR
The work tackles proving existence and symmetry of localized stationary solutions for autonomous semilinear PDEs in $\mathbb{R}^m$ with symmetry group $\mathcal{G}$, focusing on 2D dihedral patterns in the Swift-Hohenberg equation. It constructs symmetry-restricted Hilbert spaces $H^l_{\mathcal{G}}$, expresses $\mathbb{F}(u)=\mathbb{L}u+\mathbb{G}(u)$, and uses a Newton-Kantorovich scheme around an approximate solution $u_0$ with a computable approximate inverse for $D\mathbb{F}(u_0)$ to obtain a true solution. Symmetry reduction via Fourier methods, including fundamental domain $J_{\mathrm{dom}}(\mathcal{G})$, trivial set $J_{\mathrm{triv}}(\mathcal{G})$, and reduced index set $J_{\mathrm{red}}(\mathcal{G})$, ensures the solution inherits $\mathcal{G}$-symmetry and enables construction of dihedral patterns, such as those with dihedral group $D_j$. The approach builds on prior work on unbounded_domain_cadiot and JB_symmetries, extends to 2D SH beyond radial symmetry, and provides GitHub-accessible algorithmic details for fully rigorous, computer-assisted proofs.
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 $H^l_{\mathcal{G}}$ 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.