Nonlinear Wave Motion

一类具有重要物理应用的基本非线性波动方程及其相关的非线性系统正在被研究。将得到一维和多维方程的新的解和性质。将被研究的具有物理意义的系统包括Kadomtsev-Petviashvili(KP)方程，它是Korteweg-DeVries方程的两个空间-一维推广，以及一空间-一维向量非线性薛定谔方程。我们将寻求四维自对偶方程解的新的常微分方程组的约化，并分析其解的性质。在水波中的实验和理论研究表明，周期波在深水中的调制会产生不可重复的混沌动力学，而孤子不会表现出这种行为。我们还将研究它在其他物理问题上的应用，例如光纤。应用数学研究的一个基本内容就是用数学模型来解释物理现象。通常，这样的模型会导致非线性系统，在数量惊人的情况下，会导致某些典型的方程。这项研究旨在通过精确和近似的分析方法和计算技术来了解这些基本方程的解及其性质。求解某些非线性波动方程的一种重要方法是所谓的逆散射变换。IST在概念上类似于傅里叶变换；IST使用的是直接散射和反向散射的方法，这些方法最初是由研究量子力学的物理学家和数学家开发的。IST允许人们构建各种物理问题(如非线性光学、水波、等离子体物理、晶格振动和相对论)中出现的方程的一般解。一类称为孤子的特殊解是非常稳定的局域脉冲波。孤子在许多物理应用中都很重要。PI的研究在世界范围内被广泛引用；科学网将PI列为数学领域被引用最多的人之一。

A class of fundamental nonlinear wave equations and related nonlinear systems that have important physical applications is being investigated. New solutions and properties of both one dimensional and multi-dimensional equations will be obtained. The physically significant systems that will be investigated include the Kadomtsev-Petviashvili (KP) equation, which is a two space - one time dimensional extension of the Korteweg-deVries equation and the one space - one time dimensional vector nonlinear Schroedinger equation. New ordinary differential equation reductions of the four-dimensional self-dual equations will be pursued and properties of their solutions analyzed. Experimental and theoretical research in water waves has shown that modulation of periodic waves in deep water yield nonrepeatible chaotic dynamics, whereas solitons do not exhibit this behavior. Applications to other physical problems, such as fiber optics, will be investigated. It appears that this phenomenon is universal in character.An essential element in the study of Applied Mathematics is to explain physical phenomena by mathematical models. Frequently such models lead to nonlinear systems and in a surprisingly large number of cases certain prototypical equations. This research aims to understand by exact and approximate methods of an analysis and computational techniques solutions to these underlying equations and their properties. An important method used to solve certain nonlinear wave equations is the so-called Inverse Scattering Transform (IST). The IST is conceptually analogous to the Fourier Transform; IST employs methods of direct and inverse scattering that are techniques originally developed by physicists and mathematicians studying quantum mechanics. The IST allows one to construct general solutions to equations that arise in a variety of physical problems such as nonlinear optics, water waves, plasma physics, lattice vibrations, and relativity. A special class of solutions referred to as solitons are extremely stable localized pulse-like waves. Solitons are important in many physical applications. The PI's research has been extensively referenced worldwide; the Web of Science lists the PI as one of the most highly cited people in the field of mathematics.