Modulations of Periodic Waves in Applied Mathematics

应用数学中的周期波调制

• 批准号: 2108749
• 负责人: Mathew Johnson
• 金额: $ 19.8万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108749&HistoricalAwards=false
• 关键词: Modulations; Periodic; Waves; Applied; Mathematics; Modulations; Periodic; Waves; Applied; Mathematics

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.

该项目分析了用于粘性流体动力学和光学信号传播模型的空间周期性解决方案的存在和行为，并着重于这种模式的稳定性，这是它们在受到小扰动时持续存在的能力。 对模式的稳定性的研究具有实际利益，因为在自然界中只能观察到稳定的解决方案。 这项研究的结果将为科学家提供新的分析工具和方法，以了解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理，这些原理将用于更好地解释实验中观察到的模式，以预测未先前观察到的新结构和模式的存在，并洞悉模式的实验性结构。 本科生和研究生都将通过研究参与该项目接受培训。研究项目自然会根据其根本不同的物理应用分为两组问题。 第一组旨在开发和分析与浮力粘性界面波动力学有关的新数学模型，例如通过多孔岩石升起的岩浆。 预计该分析的结果将为目前无法解释的实验室观察提供数学上严格的理由。 第二个涉及光学波导和光学谐振器的动力学，并将解决实验学家和理论家都感兴趣的几个出色问题。 重要的是，这两个组件均涉及数学工作，可以在实验测试中与实验室实验进行比较。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。