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

这项合作建议涉及的非线性，色散波现象的Kadomtsev-Petviashvili（KP）方程及其物理应用，特别是在二维浅水波的调查。KP方程存在一个称为线孤子的特殊解，它是一种高振幅的稳定传播波，类似于海滩波。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）开发一种渐近理论，其中KP方程作为二维波动现象中真实的应用的首阶方程。本计画将对KP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。理论结果将与实验测量值仔细比较。初步工作表明，这些新发现的KP方程的解决方案中的一些可能有重要的物理应用，如马赫反射的斜入射波到一个垂直的墙壁和在产生大振幅的“流氓”波在浅水附近的海滩。本研究的目的之一是应用该项目的结果，以调查在公海和沿着海岸线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。拟议的研究活动将涉及几个本科生和研究生谁将获得第一手的应用数学研究经验。由于该理论被各种其他物理系统所共享，预计拟议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.