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

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特性，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了解在公海和海岸线上经常观察到的产生极高海浪的可能机制。这种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，特别是在人口密集的沿海地区附近的海洋，是一项重要而紧迫的任务。研究活动将涉及本科生和研究生，他们将在应用数学领域接受培训，并获得第一手的研究经验。本文的研究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网型的孤立波解。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。从广义上讲，本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，例如在浅水中，KP方程描述了阶模型；(2)利用几何和组合理论研究KP网络孤子复杂二维模式的详细结构，这为非线性波领域带来了新的技术。特别是，KP网孤子的相互作用、稳定性和初值问题的详细分析和数值研究将在本项目中进行。此外，该项目设想与实验人员合作，以便将获得的理论结果与在水波箱中进行的实验室实验进行比较。

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.