Collaborative Research: Evolution Systems On Time-Dependent Domains: Study Of Dynamics, Stability, And Coarsening

这一合作研究项目的目标是理解和量化在时间固定和时间依赖的空间域上扩展系统的结构形成和演化之间的关键差异。要实现这些目标，需要在理论工具和实验能力协同作用的基础上开发新的方法，并将导致动力分叉理论的新进展。这项研究将导致描述依赖于时间的区域上的图案波长的非线性演化的新模型，以便与在固定区域上的模型进行比较，包括通过Eck haus不稳定性产生相移。将与时间相关的区域上的图案粗化与固定区域上的图案粗化进行对比。为了验证理论并为理论思想和结果提供实验平台，我们将在依赖于时间的区域上对法拉第波进行广泛的实验研究。从输运反应过程-晶体生长、金属铸造、气液和气固反应系统到经典的具有动壁的电磁谐振腔、量子力学问题、流体运动、流固相互作用以及生物中结构和形态的形成等等，许多自然系统都在依赖于时间的空间域上演化。因此，从基本和实用的角度揭示时变域上的动力学是很重要的。在技术层面，该计划的成果将提供对扩展系统在依赖时间的领域的行为的基本见解，这反过来将导致上述应用中结构预测和控制的进步。在传播层面，所研究的现象为以不同程度的复杂性向人们介绍科学技术和鼓励科学培训提供了一个很好的机会。研究生、本科生和地区性高中生不仅将从课堂教学中受益，还将获得第一手研究经验。

The goal of this collaborative research project is to understand and quantify the key differences between structure formation and evolution in extended systems on time-fixed and time-dependent spatial domains. Achieving these objectives will require the development of new methods based on the synergy of theoretical tools and experimental capabilities, and will lead to new advances in dynamical bifurcation theory. The study will lead to new models describing the nonlinear evolution of pattern wavelength on time-dependent domains for comparison with those on fixed domains, including the generation of phase slips via Eckhaus instability. Pattern coarsening on time-dependent domains will be contrasted with that on fixed domains. To validate the theory and provide a testbed for theoretical ideas and findings, an extensive experimental study of Faraday waves on time-dependent domains will be performed.There are many natural systems which evolve on time-dependent spatial domains ranging from transport-reaction processes - crystal growth, metal casting, gas-liquid and gas-solid reaction systems - to classical electromagnetic cavity resonators with moving walls, quantum mechanical problems, fluid motion, fluid-structure interaction, and formation of structure and morphology in biology, to name but a few. Therefore, revealing the dynamics on time-dependent domains is important both from fundamental and practical points of view. At the technological level, the outcome of this program will provide fundamental insights into the behavior of extended systems on time-dependent domains, which in turn should lead to advances in structure prediction and control in the above-mentioned applications. At the dissemination level, the studied phenomena present a great opportunity to introduce people to science and technology at varying degrees of complexity and to encourage scientific training. Graduate, undergraduate, and regional high-school students will benefit not only through classroom instruction, but will also be provided with firsthand research experience.