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

Ian Melbourne, Jens Rademacher, Bob Rink, Sergey Zelik

TL;DR

This work develops a general mechanism for the existence and basic properties of quasicrystals in spatially extended systems with Euclidean symmetry through spontaneous symmetry breaking. It constructs flow-invariant subspaces hosting quasicrystal solutions via holohedries, extends planar and higher-dimensional quasicrystals, and establishes local (and partial global) well-posedness within $\ell^1({\mathcal{L}}^*_H)$ and related spaces, avoiding heavy implicit-function theorems. The Swift-Hohenberg equation is treated in detail, yielding global existence in low dimensions and the explicit $\|u_\lambda(t)\|_2 \sim C\sqrt{\lambda}$ scaling for the diffraction-norm, along with asymptotic persistence results showing energy concentration near critical frequencies. The analysis also discusses Turing-type instabilities in reaction-diffusion systems and highlights how the quasicrystal framework extends beyond purely periodic patterns, providing a foundation for Part II on global attractors and long-time behavior. Overall, the paper offers a robust, symmetry-driven pathway to predict and analyze quasicrystal-like patterns in complex PDEs, with implications for pattern formation in fluids, materials, and beyond.

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.

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

TL;DR

This work develops a general mechanism for the existence and basic properties of quasicrystals in spatially extended systems with Euclidean symmetry through spontaneous symmetry breaking. It constructs flow-invariant subspaces hosting quasicrystal solutions via holohedries, extends planar and higher-dimensional quasicrystals, and establishes local (and partial global) well-posedness within and related spaces, avoiding heavy implicit-function theorems. The Swift-Hohenberg equation is treated in detail, yielding global existence in low dimensions and the explicit scaling for the diffraction-norm, along with asymptotic persistence results showing energy concentration near critical frequencies. The analysis also discusses Turing-type instabilities in reaction-diffusion systems and highlights how the quasicrystal framework extends beyond purely periodic patterns, providing a foundation for Part II on global attractors and long-time behavior. Overall, the paper offers a robust, symmetry-driven pathway to predict and analyze quasicrystal-like patterns in complex PDEs, with implications for pattern formation in fluids, materials, and beyond.

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 norm of the diffraction diagram grows like the square root of the bifurcation parameter.

Paper Structure

This paper contains 11 sections, 8 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Comparison with mathematical quasicrystals
  3. Existence of quasicrystals via spontaneous symmetry breaking
  4. Flow-invariant subspaces of quasicrystals
  5. Planar quasicrystals with holohedry ${\mathbb D}_q$
  6. Higher-dimensional quasicrystals
  7. Local well-posedness and local existence of quasicrystal solutions
  8. Global existence of quasicrystal solutions
  9. The Swift-Hohenberg equation
  10. Asymptotic persistence of quasicrystal structure
  11. Reaction diffusion systems and Turing instabilities

Key Result

Proposition 2.4

Let $u\in \ell^1({\mathcal{L}}^*_H)$ as above. Then $u$ is $H$-invariant if and only if $a_{\gamma k}=a_k$ for all $\gamma\in H$, $k\in{\mathcal{L}}^*_H$.

Figures (1)

  • Figure 1: Experimental evidence of quasicrystal structures. (a) Electron diffraction diagram of a metallic solid. See Shechtman et al.SBGC for details. Reprinted Fig. 2 with permission from SBGC. Copyright (1984) by the American Physical Society. (b) Photograph of a Faraday wave experiment. See Edwards & Fauve EF for details. Reprinted Fig. 3 with permission from EF. Copyright (1993) by the American Physical Society.

Theorems & Definitions (24)

  • Remark 1.1
  • Remark 1.2
  • Remark 2.1
  • Remark 2.2: Superquasicrystals
  • Remark 2.3: Anti-quasicrystals
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Corollary 2.7
  • Remark 2.8
  • ...and 14 more