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了解和量化的时间固定和时间依赖的空间域上的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开发新的方法的基础上的理论工具和实验能力的协同作用，并将导致新的进展动态分岔理论。这项研究将导致新的模型描述的模式波长的非线性演化的时间依赖域与固定域的比较，包括通过埃克豪斯不稳定性的相位滑移的产生。将时间依赖域上的模式粗化与固定域上的模式粗化进行对比。为了验证该理论并为理论思想和发现提供试验平台，将进行时间相关域上的法拉第波的广泛实验研究。有许多自然系统在时间相关的空间域上演化，从传输-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、流体-结构相互作用以及生物学中的结构和形态的形成，仅举几例。因此，揭示含时域上的动力学从基础和实际的角度来看都是重要的。在技术层面上，该计划的成果将为扩展系统在时间依赖域上的行为提供基本见解，这反过来又会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，本科生和地区高中学生将受益不仅通过课堂教学，但也将提供第一手的研究经验。

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.