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Modulations of Periodic Waves in Applied Mathematics

应用数学中的周期波调制

基本信息

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

项目摘要

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.

该项目分析了粘性流体动力学和光信号传播模型的空间周期解的存在性和行为，重点是这种模式的稳定性，即它们在受到小扰动时持续存在的能力。模式稳定性的研究具有实际意义，因为在自然界中只有稳定的解才能被观察到。这项研究的结果将为科学家提供新的分析工具和方法，以了解在流体动力学和光信号传播背景下观察到的实验室实验，并将导致实验设计原则，用于更好地解释实验中观察到的模式，预测新的结构和模式的存在以前没有观察到，并提供深入了解模式的实验建设。本科生和研究生都将通过参与项目的研究来接受培训。研究项目根据其根本不同的物理应用自然分为两组问题。第一组旨在开发和分析与浮力驱动的粘性界面波动力学有关的新数学模型，例如岩浆通过多孔岩石上升。预计该分析的结果将为目前无法解释的实验室观察结果提供严格的数学依据。第二个涉及到光学波导和光学谐振腔的动力学，并将解决几个悬而未决的问题，是感兴趣的实验和理论家一样。重要的是，这两个组成部分涉及数学工作，可以通过实验测试和实验室实验进行比较。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。