论文标题
标态抛物线方程中激发扭结的弱和强相互作用
Weak and strong interaction of excitation kinks in scalar parabolic equations
论文作者
论文摘要
由Greenberg-Hastings细胞自动机(GHCA)作为可激发系统的漫画的研究,在本文中,我们研究了标量反应扩散型$θ$ Equ-Equ-Equ-Equ-Equ-Equ-Equ-Equ-Equiates gitable Angular相相动力学的可激发介质的最简单的PDE模型。一方面,我们使用比较原理和梯田理论对几何扭结位置进行定性研究。这将产生最小的初始距离作为全局下限,这是扭结和反京链的碰撞数据的明确定义的序列,并暗示周期性的纯扭结序列是渐近的等距。另一方面,我们利用弱相互作用理论来研究某些分析性扭结位置的有限扭结的亚稳态动力学,这承认对ODE的严格减少。通过爆破型奇异重新缩放,我们表明距离在有限的时间内被订购,并最终分歧。我们得出的结论是,扩散意味着有关扭结距离的信息丢失,因此基于GHCA中的位置和碰撞的熵复杂性并不简单地延续到PDE模型中。
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.