Nonlinear Waves in Fluids

流体，如空气或水，支持许多不同种类的波。例如，声波在可压缩流体中传播，水波在海洋表面传播，大尺度大气波影响地球的天气和气候。在足够低的强度下，波是用线性方程组来模拟的，这是相对容易理解的；但是在更高的强度下，方程变成非线性的。非线性引入了基本的数学难题，并导致了新的物理现象，例如超音速飞机产生的冲击波，在浅水上传播的孤子（粒子状波），以及产生附加波的非线性波相互作用。该项目的目的是研究流体中的波动力学，并在特定物理应用的背景下揭示新的非线性现象。该奖项还将为参与该项目的一名研究生提供支持。该项目涉及各种流体系统中非线性波传播的数学建模和分析。它着重于非线性、非色散和色散波的渐近分析，包括在边界、自由表面或界面上传播的波。哈密顿动力学提供了一个统一的框架，它是拟议研究中所考虑的波的基本结构。物理应用包括不可压缩流体中的涡度不连续波、表面准地转流中的温度锋、等离子体中的电流涡片、激波反射以及声音和熵波的共振相互作用。许多这样的问题导致了共同的数学主题，如退化色散冲击，准线性波动方程的范式变换，以及奇点的形成和传播。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Fluids, such as air or water, support many different kinds of waves. For example, sound waves propagate through compressible fluids, water waves propagate on the surface of an ocean, and large-scale atmospheric waves affect the earth's weather and climate. At sufficiently low intensities, waves are modeled by linear systems of equations, which are relatively well-understood; but at higher intensities, the equations become nonlinear. Nonlinearity introduces fundamental mathematical difficulties and leads to new physical phenomena, such as the shock waves generated by supersonic aircraft, the solitons (particle-like waves) that propagate on shallow water, and nonlinear wave interactions that generate additional waves. The aim of this project is to study the dynamics of waves in fluids and uncover new nonlinear phenomena in the context of specific physical applications. This award will also provide support for the involvement of one graduate student in the project. This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of fluid systems. It focuses on the asymptotic analysis of nonlinear, nondispersive and dispersive waves, including waves that propagate on boundaries, free surfaces, or interfaces. A unifying framework is provided by Hamiltonian dynamics, which is an essential structure for the waves considered in the proposed research. Physical applications include waves on vorticity discontinuities in incompressible fluids, temperature fronts in surface quasi-geostrophic flows, current-vortex sheets in plasmas, shock wave reflection, and the resonant interaction of sound and entropy waves. A number of these problems lead to common mathematical themes, such as degenerate dispersive shocks, normal form transformations for quasilinear wave equations, and the formation and propagation of singularities.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.