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.
