Dispersive Hydrodynamics and Applications

海洋中的海啸和强激光光纤都是波的传播速度取决于波的波长和幅度/强度的例子。在支持波的介质中，这两种特性的重合不仅限于海洋和纤维，而且在自然界、实验室和技术中普遍存在。有一个新兴的应用数学领域，称为色散流体力学，它探索这两个性质的数学和物理分支。该项目将开发各种物理环境中的非线性色散波的新的数学解和模型。解将包括孤子或孤波和色散冲击波，这是波的典型表现形式，其速度取决于波的幅度和波长。还将建立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。因此，这个项目是真正跨学科的。色散流体动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。小尺度孤子/孤波和大尺度水动力波(频散激波、稀疏波)都是非线性色散波动方程的解。该项目将可积和不可积色散偏微分方程组、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研究结合在一起。虽然孤子和色散激波的数学已经孤立地得到了广泛的发展，但这个项目探索了与色散流体动力流中孤子传播有关的丰富的和物理动机的新数学问题集。孤子和色散流体动力流动所固有的尺度分离使得它们之间相互作用的渐近描述成为一种方便而普遍的形式，用一个带有线性退化场的双曲型偏微分方程组来描述。将这一描述扩展并应用于非凸、多维和有吸引力的弥散流体动力流动是本项目的主要目标。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.