Modulations of Periodic Waves in Applied Mathematics

这个项目分析了粘性流体动力学和光信号传播模型的空间周期解的存在和行为，重点是这种模式的稳定性，即它们在受到小扰动时的持续能力。对模式稳定性的研究具有实际意义，因为自然界中只能观察到稳定的解。这项研究的结果将为科学家提供新的分析工具和方法，以理解在流体动力学和光学信号传播环境中观察到的实验室实验，并将导致实验设计原则，这些原则将被用于更好地解释实验中观察到的模式，预测以前没有观察到的新结构和模式的存在，并提供对模式的实验结构的洞察。本科生和研究生都将通过参与项目的研究进行培训。根据他们根本不同的物理应用，研究项目自然分为两组问题。第一组试图开发和分析与浮力驱动的粘性界面波动力学有关的新的数学模型，例如岩浆从多孔岩石中上升。这一分析的结果预计将为目前无法解释的实验室观察提供严格的数学理由。第二个涉及光学、波导和光学谐振器的动力学，将解决实验者和理论家都感兴趣的几个悬而未决的问题。重要的是，这两个部分都涉及数学工作，可以进行实验测试并与实验室实验进行比较。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project analyzes the existence and behavior of spatially periodic solutions to models for viscous fluid dynamics and optical signal propagation, with an emphasis on the stability of such patterns, that is their ability to persist when subjected to small perturbations. The study of the stability of a pattern is of practical interest since only stable solutions are expected to be observed in nature. Results of this research will provide scientists with new analytical tools and methodologies to understand observed laboratory experiments in both fluid dynamics and optical signal propagation contexts, and will lead to experimental design principles that will be used to better explain patterns observed in experiments, to predict the presence of new structures and patterns not previously observed, and to provide insight into the experimental construction of patterns. Both undergraduate and graduate students will receive training through research involvement in the project.The research project naturally divides into two sets of questions according to their fundamentally different physical applications. The first set seeks to develop and analyze new mathematical models relating to buoyancy driven viscous interfacial wave dynamics, such as magma rising through a porous rock. The results of this analysis is expected to provide mathematically rigorous justifications for currently unexplained laboratory observations. The second relates to the dynamics of optics waveguides and optical resonators and will resolve several outstanding issues that are of interest to both experimentalists and theoreticians alike. Importantly, both components involve mathematical work that can be experimentally tested and compared to laboratory experiments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.