Quasicrystals in pattern formation, Part I: Local existence and basic properties

TL;DR

This paper introduces a universal mechanism for the existence of quasicrystals in spatially extended systems, explaining their symmetry properties and diffraction features through spontaneous symmetry breaking, with detailed analysis for the Swift-Hohenberg equation.

Contribution

It proposes a general, symmetry-breaking-based approach to quasicrystal existence in PDE systems, bypassing complex mathematical conditions.

Findings

Quasicrystals with higher rotational symmetry naturally emerge from symmetry breaking.

Diffraction diagrams of these solutions resemble those of natural quasicrystals.

In the Swift-Hohenberg equation, diffraction norm scales with the square root of the bifurcation parameter.

Abstract

In this paper, we propose a general mechanism for the existence of quasicrystals in spatially extended systems (partial differential equations with Euclidean symmetry). We argue that the existence of quasicrystals with higher order rotational symmetry, icosahedral symmetry, etc, is a natural and universal consequence of spontaneous symmetry breaking, bypassing technical issues such as Diophantine properties and hard implicit function theorems. The diffraction diagrams associated with these quasicrystal solutions are not Delone sets, so strictly speaking they do not conform to the definition of a ``mathematical quasicrystal''. But they do appear to capture very well the features of the diffraction diagrams of quasicrystals observed in nature. For the Swift-Hohenberg equation, we obtain more detailed information, including that the $\ell^2$ norm of the diffraction diagram grows like the square root of the bifurcation parameter.