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Dispersive Hydrodynamics and Applications

分散流体动力学及其应用

基本信息

批准号：
1816934
负责人：
Mark Hoefer
金额：
$ 39.88万
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：
2018
资助国家：
美国
起止时间：
2018-07-15 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1816934&HistoricalAwards=false
关键词：
Dispersive Hydrodynamics Applications

项目摘要

Tsunamis in the ocean and intense laser fiber optics are examples of waves whose speed of propagation depend on both the wave's wavelength and amplitude/intensity. The coincidence of these two properties in media that support waves is not limited to the ocean and fibers but is ubiquitous in nature, the lab, and technology. There is an emerging field of applied mathematics called dispersive hydrodynamics that explores the mathematical and physical ramifications of these two properties. This project will develop new mathematical solutions and models of nonlinear, dispersive waves in a variety of physical environments. The solutions will include solitons or solitary waves and dispersive shock waves, quintessential manifestations of waves whose speeds depend on wave amplitude and wavelength. An in-house laboratory shallow water experiment will also be constructed to realize and test the physical implications of the developed mathematics. Additional applications to fluid dynamics, nonlinear optics, and quantum fluids will be by-products of this research. Consequently, this project is truly interdisciplinary.Dispersive hydrodynamics has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media. Small-scale solitons/solitary waves and large-scale hydrodynamic waves (dispersive shock waves, rarefaction waves) are both solutions of nonlinear dispersive wave equations. This project weaves together research on integrable and nonintegrable dispersive partial differential equations, convex and nonconvex dispersive hydrodynamic systems, multidimensional waves, asymptotic analysis, and applications that include an in-house shallow water experiment. While the mathematics of solitons and that of dispersive shock waves have been extensively developed in isolation, this project explores the rich and physically motivated set of new mathematical problems related to soliton propagation in dispersive hydrodynamic flows. The scale separation inherent to solitons and dispersive hydrodynamic flows enables the asymptotic description of their interaction in a convenient and universal form in terms of a hyperbolic system of partial differential equations with a linearly degenerate field. Extending and applying this description to nonconvex, multidimensional, and attractive dispersive hydrodynamic flows are the primary aims of this project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

海洋中的海啸和强激光光纤是波的例子，其传播速度取决于波的波长和振幅/强度。这两种特性在支持波的介质中的重合不仅限于海洋和纤维，而且在自然界、实验室和技术中无处不在。有一个新兴的应用数学领域称为色散流体力学，探讨这两个属性的数学和物理分支。该项目将开发各种物理环境中非线性色散波的新数学解决方案和模型。这些解将包括孤立子或孤立波和色散激波，它们是波的典型表现形式，其速度取决于波的振幅和波长。还将建造一个内部实验室浅水实验，以实现和测试所开发的数学的物理含义。流体动力学、非线性光学和量子流体的其他应用将是这项研究的副产品。色散流体力学已经成为描述色散介质中多尺度非线性波动现象的统一数学框架。小尺度孤立子/孤立波和大尺度流体力学波（色散激波、稀疏波）都是非线性色散波方程的解。该项目将可积和不可积的色散偏微分方程，凸和非凸色散流体动力学系统，多维波，渐近分析以及包括内部浅水实验在内的应用程序的研究编织在一起。虽然孤立子的数学和色散冲击波的数学已经被广泛地孤立地发展，但本项目探索了与色散流体动力学流中孤立子传播相关的丰富和物理动机的新数学问题。尺度分离固有的孤子和色散流体动力学流，使渐近描述他们的相互作用在一个方便和普遍的形式，在双曲型系统的偏微分方程与线性退化场。扩展和应用这种描述非凸，多维，和有吸引力的分散流体动力学流是这个项目的主要目标。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。