Quasicrystals in pattern formation. Part II: Spatially almost-periodic profiles and global existence

TL;DR

This paper develops a general framework for constructing time-evolving quasicrystals in spatially-extended PDE systems, emphasizing almost-periodic functions and global solutions, with applications to models like Swift-Hohenberg and Brusselator equations.

Contribution

It introduces a new mechanism for globally existing, spatially almost-periodic quasicrystals that avoids complex Diophantine conditions and implicit function theorems.

Findings

Constructed quasicrystals with rotational and icosahedral symmetry.

Demonstrated global existence of solutions in non-decaying function classes.

Applied the approach to Swift-Hohenberg and Brusselator models.

Abstract

This paper continues our study of quasicrystals initiated in Part I. We propose a general mechanism for constructing quasicrystals, existing globally in time, in spatially-extended systems (partial differential equations with Euclidean symmetry) and demonstrate it on model examples of the Swift-Hohenberg and Brusselator equations. In contrast to Part I, our approach here emphasises the theory of almost-periodic functions as well as the global solvability of the corresponding equations in classes of spatially non-decaying functions. We note that the existence of such time-evolving quasicrystals with rotational symmetry of all orders, icosahedral symmetry, etc., does not require technical issues such as Diophantine properties and hard implicit function theorems, which look unavoidable in the case of steady-state quasicrystals. This paper can be largely read independently of Part I. Background material and definitions are repeated for convenience, but some elementary calculations from Part I are omitted.