Nonlinear Wave Motion

研究将集中在构造的解决方案和一类物理上重要的非线性波动方程的性质的调查。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。从以前的NSF支持的研究结果的PI，它是已知的，新的多块孤子可以出现在水波具有相对较大的表面张力。在这种情况下，将考虑多块类解决方案的完整表征。也将研究具有类似类型解的相关非线性方程。新的类的非局部方程，可解的逆散射变换将进行分析。 在各种各样的物理上感兴趣的系统中的色散激波的相互作用将被考虑的逆散射变换。本课程也将研究具有周期性晶格背景的光学材料中的非线性波传播。海洋和光学中的波动现象具有广泛的吸引力。PI和NSF资助的一名研究生对平坦海滩上浅水波相互作用进行了大量观察，并得出了数学描述，为此类现象提供了更深入的理解。这些相互作用经常在视觉上表现为X和Y结构，尽管有时它们更复杂。有趣的是，这些波也可以应用于海啸传播。 作为初步研究的结果，最近在《物理评论》上发表了一篇期刊文章。这项工作随后在《今日物理》、《工业与应用数学新闻学会》的一篇焦点文章中进行了描述，并在一些受欢迎的新闻机构的文章中进行了专题报道。这些文章包括作者拍摄的精彩照片。在其他方面，数学也适用于非线性光学，其应用涉及局部电磁波的动力学，转向和操纵。预期的研究将扩展非线性方程的数学理解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.