有限尺寸系统中稳定非线性条纹图的带宽
On the band-width of stable nonlinear stripe patterns in finite size systems
摘要：
非线性条纹图案出现在许多不同的系统中，从生物细胞的小尺度到云图案的地质尺度。它们都具有在不同波数$q$下稳定的普适性质，即，它们是多稳态的。条纹图案的稳定波数范围，这是有限的Eckhaus和锯齿形的不稳定性，即使在有限的系统中的几个边界条件，随着系统尺寸的减小而增加。这种放大是因为抑制自由度从两个不稳定性去随着系统的减少，和放大取决于边界条件，我们显示分析和数值与通用的Swift-Hohenberg（SH）模型和通用的Newell-Whitehead-Segel方程。我们还描述了如何，在非常小的系统尺寸，从基本状态出现的任何周期性图案是同时稳定的在一定的参数范围内，这是特别重要的图灵模式的细胞。此外，我们解释了为什么低于一定的系统宽度的条纹图案的行为准一维的二维系统。此外，我们与SH模型在中型矩形域的数值模拟如何不稳定的条纹图案演变通过锯齿形不稳定性不同的边界条件的不同组合成稳定的图案。
英文摘要：
Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers $q$, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for Turing pattern in cells. In addition, we explain why below a certain system width stripe pattern behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.
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