Nonlinear Waves in Fluids

流体，如空气或水，支持许多不同种类的波。例如，声波通过可压缩流体传播，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，波由线性方程组建模，这是相对较好理解的;但在较高的强度下，方程变得非线性。非线性引入了基本的数学困难，并导致新的物理现象，如超音速飞机产生的冲击波，在浅水中传播的孤子（粒子波），以及产生额外波的非线性波相互作用。该项目的目的是研究流体中波动的动力学，并在特定的物理应用中发现新的非线性现象。该奖项还将为一名研究生参与该项目提供支持。本计画主要研究非线性波在各种流体系统中传播的数学模型与分析。它侧重于非线性，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。一个统一的框架提供了哈密顿动力学，这是一个必要的结构，在拟议的研究中考虑的波。物理应用包括不可压缩流体中涡度不连续波、表面准地转流中的温度锋、等离子体中的电流涡面、激波反射以及声波和熵波的共振相互作用。这些问题中的一些导致了共同的数学主题，如退化色散冲击，准线性波动方程的正规形式变换，以及奇异点的形成和传播。该奖项反映了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。

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.