Nonlinear Wave Motion

物理现象，如波的传播，通常用非线性方程来表示。这种方程可以描述大振幅行为和大振幅波动。虽然很难找到大多数非线性方程的解，但有一类方程具有深厚的数学结构，并允许使用一组重要的特殊波解，称为“孤子”。孤子是物理学家和工程师非常感兴趣的局域稳定波。它们出现在不同的领域，如非线性光学、流体动力学、玻色-爱因斯坦凝聚、磁系统和等离子体物理等。在这个项目中，我们将研究一类物理上重要的非线性孤子方程的新解和性质。这项研究将研究多维方程的新解和性质，包括Kadomtsev-Petviashvili方程的多集合解；研究可以用模形式表示的非线性方程，包括自对偶杨-Mills系统的新约化；研究连续和离散的标量和矢量非线性薛定谔方程的边值问题；以及研究玻色-爱因斯坦凝聚和非线性光学中的色散激波现象。最近在玻色-爱因斯坦凝聚和非线性光学中的实验和理论研究展示了新的波动现象。在适当的条件下，观察到了一个窄的孤子波前和一个相关的调制波列。这些实验说明了所谓的频散爆炸波或频散冲击波。这种波现象是色散的、非耗散的、相当于众所周知的大气爆炸波的波，而大气爆炸波本质上是耗散的。在这个项目中，将研究一些与色散激波相关的新的基础性研究方向，包括色散激波和色散稀疏波的相互作用。

Physical phenomena such as wave propagation are often represented mathematically by nonlinear equations. Such equations can describe large amplitude behavior and large amplitude wave motion. While it is difficult to find solutions to most nonlinear equations, there is a subclass of equations that have deep mathematical structure and admit an important set of special wave solutions termed "solitons." Solitons are localized, stable waves that are of keen interest to physicists and engineers. They arise in diverse fields such as nonlinear optics, fluid dynamics, Bose-Einstein condensation, magnetic systems, and plasma physics, amongst many others. In this project new solutions and properties of a class of physically important nonlinear soliton equations will be investigated. The research will study new solutions and properties of multidimensional equations, including multi-lump solutions to the Kadomtsev-Petviashvili equation; investigate nonlinear equations that can be expressed in terms of modular forms, including new reductions of the self-dual Yang-Mills system; investigate boundary value problems for continuous and discrete scalar and vector nonlinear Schrodinger equations; and study dispersive shock wave phenomena in Bose-Einstein condensation and nonlinear optics.Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics has demonstrated novel wave phenomena. Under suitable conditions a narrow soliton wave front with an associated modulated wave train are observed. The experiments illustrate what is termed dispersive blast waves or dispersive shock waves. This wave phenomenon is the dispersive, non-dissipative, wave equivalent of well-known atmospheric blast waves which are dissipative in nature. In this project a number of new and fundamental research directions associated with dispersive shock waves will be studied, including interactions of dispersive shock waves and dispersive rarefaction waves.