Nonlinear Wave Motion

研究将集中在解的构造和研究一类物理上重要的非线性波动方程的性质。应用包括水波和非线性光学。在水波中，当表面张力相对较小时，要研究的解描述具有X， Y和更复杂结构的非衰减孤子波的相互作用。从先前由PI支持的NSF研究结果来看，已知新的多块孤子可以出现在具有相对较大表面张力的水波中。在这种情况下，将考虑多块类解的完整表征。具有类似解的相关非线性方程也将被研究。本文将分析可由逆散射变换求解的一类新的非局部方程。色散激波在各种各样的物理系统中的相互作用将通过逆散射变换来考虑。非线性波在具有周期性晶格背景的光学材料中的传播也将被研究。海洋和光学中的波浪现象具有广泛的吸引力。由国家科学基金会资助的PI和研究生对平坦海滩上浅水波浪相互作用的大量观察导致了对这种现象的更深层次理解的数学描述。这些相互作用通常在视觉上表现为X和Y结构，尽管有时它们更复杂。有趣的是，这些波浪也适用于海啸的传播。作为初步研究的结果，一篇期刊文章最近发表在《物理评论》上。这项工作随后在《今日物理》、《工业和应用数学新闻学会》的一篇重点文章中进行了描述，并在一些流行新闻机构的文章中进行了介绍。这些文章包括作者拍摄的引人注目的照片。在其他方向，数学也适用于非线性光学的应用，涉及动力学，转向和局部电磁波的操作。预期的研究将把非线性方程的数学理解扩展到各种物理上有趣的系统。pi的工作已被世界各地的研究人员广泛引用和使用。

Research will center on the construction of solutions and the investigation of the properties of a class of physically significant nonlinear wave equations. Applications include water waves and nonlinear optics. In water waves, when surface tension is relatively small, the solutions to be investigated describe interactions of non-decaying soliton waves that have X, Y and more complex structure. From results of prior NSF supported research by the PI, it is known that novel multi-lump solitons can occur in water waves with relatively large surface tension. In this case a complete characterization of the multi-lump class of solutions will be considered. Related nonlinear equations that have similar types of solutions will also be investigated. New classes of nonlocal equations that are solvable by the inverse scattering transform will be analyzed. Interactions of dispersive shock waves in a wide variety of physically interesting systems will be considered by the inverse scattering transform. Nonlinear wave propagation in optical materials with periodic lattice backgrounds will also be studied.Wave phenomena in oceans and optics have broad appeal. Numerous observations of shallow water wave interactions on flat beaches by the PI and a graduate student funded by NSF have led to mathematical descriptions that provide deeper understanding of such phenomena. These interactions frequently appear visually to be of X and Y structure, though sometimes they are more complex. Interestingly these waves also have application to tsunami propagation. As a result of the initial research, a journal article was recently published in Physical Review. The work was subsequently described in a focus article in Physics Today, the Society of Industrial and Applied Math News and was featured in a number of articles by popular news organizations. These articles included remarkable photos taken by the authors. In other directions, the mathematics also applies to nonlinear optics with applications that involve the dynamics, steering and manipulation of localized electromagnetic waves. The contemplated research will extend the mathematical understanding of the nonlinear equations to a variety of physically interesting systems. The PIs work has been widely referenced and used by researchers worldwide.