Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices
Patterns and Quasipatterns from the Superposition of Two Hexagonal Lattices
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DOI：
10.1137/20m1372780
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
G. Iooss;A. Rucklidge
中科院分区：
文献类型：
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作者：
G. Iooss;A. Rucklidge
具有8倍、10倍、12倍和更高旋转对称性的准模式已知存在于形成模式的Swift-Hohenberg偏微分方程的4个解中，由两个相互旋转一个角度α的等振幅六边形模式叠加而成的具有6倍旋转对称性的准模式5也是如此。本文考虑具有二次非线性和三次非线性的Swift-Hohenberg 7方程，并证明了几个新的拟8项的存在性：由六边形和几乎任意方向和任意相对平移的条纹（卷）叠加而成的拟图案，以及由10个不等振幅的六边形叠加而成的拟图案（只要二次非线性系数较小）。我们也考虑了周期情况，并扩展了已知解的类别，包括六边形和条纹的苏-12排列。我们的工作给出了一个准周期13等变分岔理论的行进方向。14
Quasipatterns with 8-fold, 10-fold, 12-fold and higher rotational symmetry are known to exist as 4 solutions of the pattern-forming Swift-Hohenberg partial differential equation, as are quasipatterns 5 with 6-fold rotational symmetry made up from the superposition of two equal-amplitude hexagonal 6 patterns rotated by an angle α with respect to each other. Here we consider the Swift-Hohenberg 7 equation with quadratic as well as cubic nonlinearities, and prove existence of several new quasipat-8 terns: quasipatterns made from the superposition of hexagons and stripes (rolls) oriented in almost 9 any direction and with any relative translation, and quasipatterns made from the superposition of 10 hexagons with unequal amplitude (provided the coefficient of the quadratic nonlinearity is small). 11 We consider the periodic case as well, and extend the class of known solutions, including the su-12 perposition of hexagons and stripes. Our work gives a direction of travel towards a quasiperiodic 13 equivariant bifurcation theory. 14