有限域上的条纹：为什么锯齿形不稳定性只是一个部分故事
Stripes on finite domains: Why the zigzag instability is only a partial story
摘要：
固定周期模式在自然科学中广泛存在，从纳米尺度的电化学和两亲系统到中尺度的流体、化学和生物介质，再到宏观尺度的植被和云模式。它们的形成通常是由于均匀状态的主要对称性破缺为条纹，通常随后是次要不稳定性，形成锯齿形和锯齿形图案。这些二次不稳定性的理想化条件下的无限域，然而，在有限域，情况是更微妙的，因为不稳定的模式也取决于边界条件。使用两个原型模型，Swift-Hohenberg方程和强迫复杂的Ginzburg-Landau方程，我们认为有界域没有通量边界条件横向的条纹，并揭示了一个独特的混合模式之间的不稳定性，位于经典之字形和Eckhaus线。这解释了条纹的稳定性在轻度曲折不稳定的制度，并在交叉的混合模式线，曲折条纹的演变在大部分的域和边界附近的缺陷的形成。结果是特别重要的大的时间尺度分离的问题，如体异质结变形在有机光伏和植被在半干旱地区，早期的时间瞬变可能发挥重要作用。
英文摘要：
Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical and biological media and to macro-scale vegetation and cloud patterns. Their formation is usually due to a primary symmetry breaking of a uniform state to stripes, often followed by secondary instabilities to form zigzag and labyrinthine patterns. These secondary instabilities are well studied under idealized conditions of an infinite domain, however, on finite domains, the situation is more subtle since the unstable modes depend also on boundary conditions. Using two prototypical models, the Swift-Hohenberg equation and the forced complex Ginzburg-Landau equation, we consider bounded domains with no flux boundary conditions transversal to the stripes, and reveal a distinct mixed-mode instability that lies in between the classical zigzag and the Eckhaus lines. This explains the stability of stripes in the mildly zigzag unstable regime, and, after crossing the mixed-mode line, the evolution of zigzag stripes in the bulk of the domain and the formation of defects near the boundaries. The results are of particular importance for problems with large time scale separation, such as bulk-heterojunction deformations in organic photovoltaic and vegetation in semi-arid regions, where early temporal transients may play an important role.
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