Response functions for drift of spiral and scroll waves

旋转螺旋波（二维）和涡卷波（三维）是在物理、化学和生物性质的许多空间扩展系统中观察到的一种自组织形式。其中最重要的是心肌，旋转波是折返性心律失常的原因，包括最致命的心室颤动。在理想条件下，螺旋/涡卷波通常围绕不移动的中心/细丝稳定地旋转。然而，实际上总是存在的任何破坏稳定性的扰动都会导致旋转频率和中心/灯丝的空间位置的逐渐变化，即漂移。理解这种漂移对应用程序至关重要。虽然在直接数值模拟中可以观察到漂移，但是这些计算通常是昂贵的并且缺乏通用性。小扰动引起的漂移存在一个普适的渐近理论。其适用性取决于所谓的响应函数（RF）的知识。在少数已知的情况下，RF基本上只在核心附近为非零。由于这种局部化，螺旋/涡卷波的行为就像点/弦对象，尽管显然是非局部的制度。这种独特的波粒二象性与螺旋波的显著稳定性直接相关。渐近理论利用这一特性，并允许，在原则上，一个更简单和数量级更有效的预测他们的漂移比直接数值模拟。一旦找到，特定模型的RF允许人们预测响应于任意扰动的螺旋和涡卷的漂移。目前的建议的目的是发展定期和通用的方法获得RF，然后使渐近理论成为一个实际工作的工具，理解和控制旋转波在真实的系统。

Rotating spiral waves (in two dimensions) and scroll waves (in three dimensions) are a form of self-organization observed in numerous spatially extended systems of physical, chemical and biological nature. The most important of these is heart muscle where rotating waves are responsible for re-entrant arrhythmias, including the most lethal one, the ventricular fibrillation. Under ideal conditions, a spiral/scroll wave commonly rotates steadily around a nonmoving center/filament. However, any symmetry-breaking perturbation, always present in reality, causes a gradual change in rotation frequency and in spatial location of the centre/filament, i.e. a drift. Understanding this drift is vitally important for applications. While drift may be observed in direct numerical simulations, these computations are often expensive and lack generality. There exists a universal asymptotic theory of drift caused by small perturbations. Its applicability is contingent on knowledge of so called response functions (RFs). In a few known cases, the RFs are essentially nonzero only near the core. As a result of this localization, spiral/scroll waves behave like point/string objects, despite being apparently nonlocal regimes. This unique kind of wave-particle duality is directly related to the remarkable stability of spiral/scroll waves. The asymptotic theory exploits this property and allows, in principle, a much simpler and orders of magnitude more efficient prediction of their drift than direct numerical simulations. Once found, RFs of a particular model allow one to predict the drift of spirals and scrolls in response to arbitrary perturbations. The current proposal aims to develop regular and generic methods of obtaining the RFs and then to make the asymptotic theory into an actually working tool for understanding and controlling rotating waves in real systems.