准模式的8、10和12模表示的收敛性质
Convergence properties of the 8, 10 and 12 mode representations of quasipatterns
摘要：
在法拉第波实验中观察到的准模式的空间傅里叶变换表明，该模式是很好地表示的8，10或12个傅里叶模式与波矢量等间隔的圆形的总和。这种表示法已被多次用作计算模式振幅对参数的弱非线性依赖性的标准微扰方法的起点。我们证明了n个这样的傅立叶模式的非线性相互作用产生了新的模式，这些模式的波矢接近原始圆的速度不快于常数乘以n^{-2}，并且有模式的组合确实达到了这个极限。在KAM理论中，小因子会给微扰理论带来困难，尽管小因子有界，但标准方法的收敛性是值得怀疑的。我们计算稳定quasipattern解决方案的三次Swift-Hohenberg方程高达33阶，以说明一些细节的问题，并认为标准的方法不收敛足够快，被视为一个可靠的方式计算quasipatures的属性。
英文摘要：
Spatial Fourier transforms of quasipatterns observed in Faraday wave experiments suggest that the patterns are well represented by the sum of 8, 10 or 12 Fourier modes with wavevectors equally spaced around a circle. This representation has been used many times as the starting point for standard perturbative methods of computing the weakly nonlinear dependence of the pattern amplitude on parameters. We show that nonlinear interactions of n such Fourier modes generate new modes with wavevectors that approach the original circle no faster than a constant times n^{-2}, and that there are combinations of modes that do achieve this limit. As in KAM theory, small divisors cause difficulties in the perturbation theory, and the convergence of the standard method is questionable in spite of the bound on the small divisors. We compute steady quasipattern solutions of the cubic Swift--Hohenberg equation up to 33rd order to illustrate the issues in some detail, and argue that the standard method does not converge sufficiently rapidly to be regarded as a reliable way of calculating properties of quasipatterns.
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