RUI: Solitary structures in nonlinear wave equations

RUI：非线性波动方程中的孤立结构

• 批准号: 0807404
• 负责人: Sarbarish Chakravarty
• 金额: $ 10.38万
• 依托单位: University of Colorado at Colorado Springs
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807404&HistoricalAwards=false
• 关键词: RUI; Solitary; structures; nonlinear; wave

The proposal concerns the investigation of nonlinear wave phenomena in shallow water waves, nonlinear optics, plasma and other areas of nonlinear physics. A comprehensive characterization of the solitary waves resulting from such phenomena is an important problem, which is far from complete. Therefore an objective of this research project is to conduct an extensive study of the nonlinear equations describing such localized wave-forms and wave-patterns by analyzing the underlying mathematical structures of the relevant solution spaces. Since the theory is shared by various physical systems, it is anticipated that the results from the proposed work would provide insights into other areas such as light waves in nonlinear optics, and spin waves in magnetic thin films. A second objective is to apply the theoretical results of this project to investigate possible mechanisms generating "rogue" waves of extremely high elevations frequently observed in open seas, and surface wave patterns in shallow water. Understanding the nature and dynamics of such extreme waves, and ultimately predicting such wave phenomena in oceans near highly populated coastal areas or at harbor entrances with heavy shipping traffic, are extremely important tasks. The proposed research activities will involve several undergraduate students over the course of this project since it will be carried out in an undergraduate institution. The students will gain firsthand research experience in applied mathematics, which may motivate them to pursue graduate studies or other career opportunities in mathematical sciences.

该提案涉及浅水波、非线性光学、等离子体和非线性物理其他领域的非线性波现象的研究。对此类现象产生的孤立波进行全面表征是一个重要问题，但还远未完成。因此，该研究项目的目标是通过分析相关解空间的基础数学结构，对描述此类局部波形和波形的非线性方程进行广泛的研究。由于该理论被各种物理系统所共享，因此预计所提出的工作的结果将为其他领域提供见解，例如非线性光学中的光波和磁性薄膜中的自旋波。第二个目标是应用该项目的理论结果来研究在公海中经常观察到的产生极高海拔“流氓”波浪以及浅水中的表面波浪模式的可能机制。了解此类极端波浪的性质和动态，并最终预测人口稠密的沿海地区附近的海洋或航运交通繁忙的港口入口处的此类波浪现象，是极其重要的任务。由于该项目将在本科机构中进行，因此拟议的研究活动将涉及多名本科生参与该项目。学生将获得应用数学的第一手研究经验，这可能会激励他们攻读数学科学领域的研究生或其他职业机会。