Nonlinear Wave Motion

非线性波动

• 批准号: 1310200
• 负责人: Mark Ablowitz
• 金额: $ 26.64万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1310200&HistoricalAwards=false
• 关键词: Nonlinear; Wave; Motion

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.

研究将集中于一类具有物理意义的非线性波动方程的解的构造和性质的研究。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，要研究的解描述了具有 X、Y 和更复杂结构的非衰减孤子波的相互作用。从 PI 先前 NSF 支持的研究结果来看，新型多团孤子可以出现在表面张力相对较大的水波中。在这种情况下，将考虑多块类解决方案的完整表征。具有相似解类型的相关非线性方程也将被研究。将分析可通过逆散射变换求解的新类非局部方程。 逆散射变换将考虑各种物理上有趣的系统中色散冲击波的相互作用。还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的浅水波相互作用进行了大量观察，得出了数学描述，可以更深入地了解此类现象。这些相互作用在视觉上经常表现为 X 和 Y 结构，尽管有时它们更复杂。有趣的是，这些波也适用于海啸传播。 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《今日物理学》、工业与应用数学新闻协会的一篇焦点文章中进行了描述，并在热门新闻机构的多篇文章中得到了专题报道。这些文章包括作者拍摄的精彩照片。在其他方向上，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。预期的研究将对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。