Quasicrystals in pattern formation. Part II: Spatially almost-periodic profiles and global existence
Ian Melbourne, Jens Rademacher, Bob Rink, Sergey Zelik
TL;DR
This work advances the study of spatially evolving quasicrystals in PDEs with Euclidean symmetry by developing a rigorous framework based on spatially almost-periodic and quasiperiodic functions and their hulls, enabling global-in-time constructions in non-decaying profiles. It provides global well-posedness results in $AP({\mathcal{L}}^*)$ and $\ell^1({\mathcal{L}}^*)$, preserves frequency modules, and derives a hull-based SH equation to control quasicrystal amplitudes. In the Swift-Hohenberg setting, the authors reveal a gradient-like structure on almost-periodic spaces and construct time-dependent quasicrystal branches on the global attractor $\mathcal{A}_\lambda$, with amplitudes scaling as $\sqrt{\lambda}$ near bifurcation, including backward-time convergence to zero. They also extend these insights to the Brusselator, proving global existence in $L^\infty_+({\mathbb R}^d)$ and demonstrating quasicrystal solutions on attractors under AP initial data, despite the lack of a global Lyapunov function. Overall, the paper generalizes Part I's ideas into a robust, implementable program for obtaining spatially non-decaying quasicrystal solutions in canonical PDE models, with implications for pattern formation and dynamics of almost-periodic structures.
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.