标量抛物型方程中激发扭结的弱相互作用和强相互作用
Weak and strong interaction of excitation kinks in scalar parabolic equations
摘要：
受Greenberg-Hastings元胞自动机（GHCA）作为可激发系统的漫画的研究的启发，在本文中，我们研究了由标量反应扩散型$θ$ -可激发角相位动力学方程给出的可激发介质的可能最简单的PDE模型中的扭结-反扭结动力学。一方面，我们利用比较原理和阶地理论定性地研究了几何纽结的位置。这产生的最小初始距离作为一个全球性的下界，一个定义良好的序列的碰撞数据扭结和反扭结，并意味着周期性的纯扭结序列是渐近等距。另一方面，我们研究了亚稳态动力学的许多扭结使用弱相互作用理论的某些解析扭结的位置，它承认一个严格的减少到常微分方程。通过爆破型奇异重标度，我们证明了距离在有限时间内变得有序，并最终发散。我们的结论是，扩散意味着一个信息的损失的扭结距离，使熵的复杂性的基础上的位置和碰撞的GHCA不简单地结转到PDE模型。
英文摘要：
Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type $θ$-equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks- and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model.
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