Collaborative research: RUI: Two-dimensional wave patterns and physical applications of Kadomtsev-Petviashvili web-solitons

合作研究：RUI：Kadomtsev-Petviashvili 网孤子的二维波型和物理应用

• 批准号: 1410862
• 负责人: Sarbarish Chakravarty
• 金额: $ 15万
• 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: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1410862&HistoricalAwards=false
• 关键词: Collaborative; research; RUI; Two; dimensional

This is a collaborative research project to investigate certain types of nonlinear wave phenomena. Due to their inherent nonlinear behavior, such waves have very important physical applications in the generation of large amplitude waves, for example in shallow water near a beach. A primary goal of this research is to provide further insights into possible mechanisms that generate extremely high waves frequently observed in open seas and along coastlines. Important examples of such phenomena include tsunamis. Understanding the nature and dynamics of such extreme waves, particularly in oceans near highly populated coastal areas, is a significant and urgent task. The research activities will involve undergraduate and graduate students who will be trained in the field of applied mathematics and will gain first-hand research experience. It is further anticipated that the results from the proposed work would be useful in the study of similar nonlinear wave phenomena that occur in other physical problems including nonlinear optics, plasmas, and spin waves in magnetic thin films.The nonlinear waves indicated above are closely described by the weakly dispersive, quasi-two-dimensional Kadomtsev-Petviashvili (KP) equation which admits a class of solitary wave solutions with complex two-dimensional web patterns. These solutions are sometimes referred to as the KP web-solitons.The purpose of this collaborative research project is to continue investigations into the physical as well as the mathematical aspects of the KP equation. Broadly speaking, this project has two main goals that are interrelated, namely, (1) to study solitary wave interactions observed in certain physical situations such as shallow water where the KP equation describes the leading order model, and (2) to investigate the detailed structure of the complex two-dimensional patterns of the KP web-solitons using geometric and combinatorial theories, which bring new techniques to the area of nonlinear waves. In particular, detailed analytical and numerical studies of the interactions, stability and initial value problem of the KP web-solitons will be conducted in this project. Furthermore, the project envisions collaboration with experimentalists in order to compare the obtained theoretical results with laboratory experiments performed in water wave tanks.

这是一个合作研究项目，旨在研究某些类型的非线性波动现象。由于其固有的非线性行为，这种波在产生大振幅波方面具有非常重要的物理应用，例如在海滩附近的浅水中。这项研究的一个主要目标是进一步深入了解在公海和海岸线经常观察到的产生极高海浪的可能机制。这类现象的重要例子包括海啸。了解这种极端海浪的性质和动态，特别是在人口稠密的沿海地区附近的海洋，是一项重大而紧迫的任务。研究活动将涉及本科生和研究生，他们将接受应用数学领域的培训，并将获得第一手研究经验。这项工作的结果将有助于研究其他物理问题中出现的类似的非线性波现象，包括非线性光学、等离子体和磁性薄膜中的自旋波。上述非线性波可以用弱色散的准二维Kadomtsev-Petviashvili(KP)方程来描述，该方程允许一类具有复杂二维腹板图案的孤立波解。这些解有时被称为KP网络孤子。这个合作研究项目的目的是继续研究KP方程的物理和数学方面。广义地说，这个项目有两个相互关联的主要目标，即(1)研究在某些物理情况下观察到的孤立波相互作用，如在浅水中观察到的KP方程描述的前序模型；(2)利用几何和组合理论研究KP网孤子复杂的二维图案的详细结构，这为非线性波领域带来了新的技术。特别是，本项目将对KP网孤子的相互作用、稳定性和初值问题进行详细的解析和数值研究。此外，该项目设想与实验者合作，以便将所获得的理论结果与在水波水箱中进行的实验室实验进行比较。