Pattern formation in the wake of triggered pushed fronts
Pattern formation in the wake of triggered pushed fronts
复制标题
触发推动前沿后形成形态
DOI：
10.1088/0951-7715/29/8/2196
复制
发表时间：
2015
期刊：
影响因子：
1.7
通讯作者：
A. Scheel
中科院分区：
文献类型：
--
作者：
Ryan N. Goh;A. Scheel
模式形成锋通常由外部刺激控制，该外部刺激以固定速度穿过稳定介质，使其在其尾流中不稳定。通过控制兴奋的速度，这种刺激或“刺激物”可以介导图案形成前沿，这些前沿自由地侵入不稳定的平衡并控制选择哪种图案。在这项工作中，我们分析和数值研究时，触发扰动的振荡推动自由锋。在这种情况下，产生的图案化的前面，我们称之为一个推动触发前，表现出各种各样的现象，包括蛇形，非单调波数选择，和滞后。假设存在一个通用的振荡推自由锋，我们使用异宿分歧技术证明存在的触发前在一个抽象的设置空间动力学方法的动机。然后，我们推导出一个领先的顺序扩展选定的波数的触发速度。此外，我们表明，这样的分叉曲线是由一定的强稳定和弱稳定的空间特征值的差异与自由推前的衰减。我们还研究了典型的例子，这些现象在五次复杂的金兹伯格朗道方程和修改后的Cahn-Hilliard方程。
Pattern-forming fronts are often controlled by an external stimulus which progresses through a stable medium at a fixed speed, rendering it unstable in its wake. By controlling the speed of excitation, such stimuli, or ‘triggers’, can mediate pattern forming fronts which freely invade an unstable equilibrium and control which pattern is selected. In this work, we analytically and numerically study when the trigger perturbs an oscillatory pushed free front. In such a situation, the resulting patterned front, which we call a pushed trigger front, exhibits a variety of phenomenon, including snaking, non-monotonic wave-number selection, and hysteresis. Assuming the existence of a generic oscillatory pushed free front, we use heteroclinic bifurcation techniques to prove the existence of trigger fronts in an abstract setting motivated by the spatial dynamics approach. We then derive a leading order expansion for the selected wave-number in terms of the trigger speed. Furthermore, we show that such a bifurcation curve is governed by the difference of certain strong-stable and weakly-stable spatial eigenvalues associated with the decay of the free pushed front. We also study prototypical examples of these phenomena in the cubic-quintic complex Ginzburg Landau equation and a modified Cahn–Hilliard equation.