Proving periodic solutions and branches in the 2D Swift Hohenberg PDE with hexagonal and triangular symmetry

TL;DR

This paper develops a rigorous method to prove the existence of smooth, periodic solutions with hexagonal and triangular symmetries in the 2D Swift-Hohenberg PDE, combining analytical and numerical Fourier series techniques.

Contribution

It introduces a novel Fourier series construction for hexagonal lattices and a Newton-Kantorovich approach to rigorously verify periodic solutions in PDEs with specific symmetries.

Findings

Proved existence of $D_3$ and $D_6$ symmetric solutions in the 2D Swift-Hohenberg PDE.

Developed a computational framework for constructing approximate solutions and bounds.

Provided algorithmic implementation details on Github.

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.