Quasicrystals in pattern formation, Part I: Local existence and basic properties
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913675229528064 |
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| author | Melbourne, Ian Rademacher, Jens Rink, Bob Zelik, Sergey |
| author_facet | Melbourne, Ian Rademacher, Jens Rink, Bob Zelik, Sergey |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19967 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quasicrystals in pattern formation, Part I: Local existence and basic properties Melbourne, Ian Rademacher, Jens Rink, Bob Zelik, Sergey Pattern Formation and Solitons Dynamical Systems 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. |
| title | Quasicrystals in pattern formation, Part I: Local existence and basic properties |
| topic | Pattern Formation and Solitons Dynamical Systems |
| url | https://arxiv.org/abs/2410.19967 |