Collaborative research: Topics related to the solitary waves of the KP equation and physical applications

该合作提案涉及Kadomtsev-Petviashvili （KP）方程描述的非线性色散波现象及其物理应用的研究，特别是在二维浅水波浪中。KP方程有一个特殊的解，叫做线孤子，它是一种稳定的高振幅传播波，就像沙滩波一样。从广义上讲，该项目有两个相互关联的主要目标，即(i)研究KP孤子解空间的组合和几何方面，以及（ii）发展一个渐进理论，其中KP方程作为二维波现象的实际应用的阶方程。本项目将对KP孤子的相互作用、稳定性和初值问题进行详细的分析和数值研究。理论结果将与实验测量结果仔细比较。初步工作表明，这些新发现的KP方程解中的一些可能具有重要的物理应用，例如斜入射波在垂直壁上的马赫反射，以及在海滩附近浅水中产生的大振幅“异常”波。这项研究的一个目标是应用该项目的结果，以调查在公海和沿海经常观察到的产生极高海拔波浪的可能机制，例如海啸。了解这种极端波浪的性质和动力学，并最终预测人口密集的沿海地区附近海洋中的这种波浪现象是重要而紧迫的任务。计划中的研究活动将涉及几名本科生和研究生，他们将获得应用数学的第一手研究经验。由于该理论被其他各种物理系统所共享，因此预计这项工作的结果将为非线性光学中的光波和磁性薄膜中的自旋波等领域提供见解。

This collaborative proposal concerns the investigation of nonlinear, dispersive wave phenomena described by the Kadomtsev-Petviashvili (KP) equation and its physical applications, particularly in two-dimensional shallow water waves. The KP equation admits a particular solution called line-soliton, which is a steady propagating wave with high amplitude, like a beach wave. Broadly speaking, the project has two main goals that are interrelated, namely, (i) to study combinatorial and geometric aspects of the solution space of the KP solitons, and (ii) to develop an asymptotic theory where the KP equation as the leading order equation for real applications in two-dimensional wave phenomena. Detailed analytical and numerical studies of the interactions, stability and initial value problem of the KP solitons will be carried out in this project. The theoretical results will be carefully compared with experimental measurements. Preliminary work suggests that some of these newly discovered solutions of the KP equation may have important physical applications such as the Mach reflection of an oblique incidence wave onto a vertical wall and in the generation of large amplitude "rogue" waves in shallow water near a beach. An objective of this research is to apply the results of this project in order to investigate possible mechanisms generating waves of extremely high elevations frequently observed in open seas and along coastlines, for example, tsunamis. Understanding the nature and dynamics of such extreme waves, and ultimately predicting such wave phenomena in oceans near highly populated coastal areas are significant and urgent tasks. The proposed research activities will involve several undergraduate and graduate students who will gain first-hand research experience in applied mathematics. Since the theory is shared by various other physical systems, it is anticipated that the results from the proposed work would provide insights into areas such as light waves in nonlinear optics, and spin waves in magnetic thin films.