Proving periodic solutions and branches in the 2D Swift Hohenberg PDE with hexagonal and triangular symmetry
Dominic Blanco
TL;DR
We address the problem of rigorously proving the existence of smooth, periodic solutions with hexagonal symmetry in PDEs, and present a Fourier-space, computer-assisted framework to construct and certify such solutions. The approach enforces space-group symmetries in Fourier series, builds approximate solutions, and applies a Newton-Kantorovich scheme with explicit bounds around an approximate inverse. The work demonstrates existence of $D_3$ and $D_6$ periodic solutions for the 2D Swift-Hohenberg PDE and extends to branches of solutions using Chebyshev series, with algorithmic details and Github availability. Key contributions include invariant hexagonal Fourier bases, explicit error bounds, and a constructive proof pipeline enabling rigorous verification of symmetric pattern solutions. The results have implications for rigorous pattern formation analysis in PDEs and provide practical tooling for computer-assisted proofs.
Abstract
In this article, we enforce space group symmetries in Fourier series to rigorously prove the existence of smooth, periodic solutions in partial differential equations (PDEs) with hexagonal and triangular symmetries. In particular, we provide the necessary analytical and numerical tools to construct Fourier series of functions on the hexagonal lattice. This allows one to build approximate solutions that are periodic. Moreover, to generate the periodic tiling, we can use one symmetric hexagon for $D_6$ symmetry and two symmetric triangles for $D_3$ symmetry. We derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, $\overline{u}$. More specifically, we verify a condition based on the computation of explicit bounds. The strategy for constructing $\overline{u}$, the approximate inverse, and the computation of these bounds will be presented. We demonstrate our approach on the 2D Swift-Hohenberg PDE by proving the existence of $D_3$ and $D_6$ periodic solutions. We then perform proofs of branches of solutions by using Chebyshev series. The algorithmic details to perform the proof can be found on Github.