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.

海洋中的海啸和强烈的激光光纤是波浪的示例，其传播速度均取决于波浪的波长和振幅/强度。支持波浪的媒体中这两个特性的巧合不仅限于海洋和纤维，而是无处不在的，本质上是实验室和技术。有一个新兴的应用数学领域，称为分散流体动力学，探讨了这两种特性的数学和物理分析。该项目将在各种物理环境中开发新的数学解决方案和非线性分散波的模型。这些解决方案将包括孤子或孤立波和分散冲击波，速度取决于波幅度和波长的波浪的典型表现。还将建立一个内部实验室浅水实验，以实现和测试已发达数学的物理意义。这项研究的副产品将用于流体动力学，非线性光学和量子流体的其他应用。因此，这个项目是真正的跨学科。分散性流体动力学已成为一个统一的数学框架，用于描述分散媒体中多尺度非线性波浪现象。小规模的孤子/孤立波和大规模流体动力波（色散冲击波，稀疏波）都是非线性色散波方程的解决方案。该项目将可集成和不可整合的分散偏微分方程，凸和非凸形分散流体动力系统，多维波，渐近分析以及包括内部浅水实验的应用进行编织。虽然孤子的数学和分散性冲击波的数学是孤立开发的，但该项目探讨了与分散流体动力流中有关孤子传播相关的丰富且具有物理动机的新数学问题。孤子和分散性流体动力流固有的尺度分离可以使其相互作用的渐近描述以方便而通用的形式，该形式以线性退化场的偏微分方程的双曲线系统来实现。将此描述扩展并应用于非概念，多维和有吸引力的分散流体动力流是该项目的主要目的。该奖项反映了NSF的法定任务，并认为值得通过基金会的智力优点和更广泛的影响审查标准通过评估来进行评估。