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

这个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中扩展系统的结构形成和演化之间的关键差异。实现这些目标将需要基于理论工具和实验能力协同作用的新方法的发展，并将导致动力分岔理论的新进展。该研究将产生新的模型，用于描述图案波长在时间依赖域上的非线性演化，并与固定域上的模型进行比较，包括通过埃克豪斯不稳定性产生的相滑移。将时间相关域上的模式粗化与固定域上的模式粗化进行对比。为了验证理论并为理论思想和发现提供一个测试平台，将对法拉第波在时间相关域上进行广泛的实验研究。有许多自然系统在依赖时间的空间域上进化，从输运反应过程-晶体生长，金属铸造，气-液和气-固反应系统-到具有移动壁的经典电磁腔谐振器，量子力学问题，流体运动，流-结构相互作用，以及生物学中结构和形态的形成，仅举几例。因此，从理论和实践的角度来看，揭示时间依赖域上的动力学是很重要的。在技术层面上，该计划的结果将为扩展系统在时间依赖域上的行为提供基本见解，这反过来将导致上述应用中结构预测和控制的进步。在传播一级，所研究的现象提供了一个很好的机会，可以向人们介绍不同复杂程度的科学和技术，并鼓励科学培训。研究生、本科生和地区高中生将不仅受益于课堂教学，还将获得第一手的研究经验。

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.