Nonlinear Surface Waves

面波是沿边界或界面传播的波。它们出现在许多物理系统中，例如海洋表面的水波和地震产生的地震面波。它们还具有广泛的技术应用，例如用于模拟滤波器的超声波表面声波和用于光学和纳米光学设备的表面等离子体。在较低强度下，不同波的效应相加(波的叠加)，可以用线性理论来描述，但在较高强度下，非线性效应变得明显。非线性导致了质的新现象，包括波的相互作用和奇点的形成(如冲击波)。本项目研究表面波的基本非线性动力学及其应用，特别是相速或频率与波长无关的非色散表面波。作为项目的一部分，研究生和本科生将通过参与研究进行培训。项目将对各种物理系统中的非线性表面波进行渐近分析。将应用于电磁表面波，或表面等离子体波，涡度不连续面波和表面准地转锋，等离子体波，和弹性中的瑞利波。由此得到的渐近方程通常在空间上是非局部的和拟线性的。我们将对这些方程进行解析和数值研究，特别是为了了解这些表面波中的非线性聚焦以及奇点可能的形成和传播。此外，该项目还将研究激波的绕射和掠过马赫反射，以及跨音速空气动力学中的相关问题。

Surface waves are waves that propagate along a boundary or interface. They arise in many physical systems, such as water waves on the surface of the ocean and seismic surface waves generated by earthquakes. They also have a wide variety of technological applications, such as ultrasonic surface acoustic waves used in analog filters and surface plasmons used in optical and nano-optical devices. At lower intensities, effects of different waves add (waves superpose) and may be described by linear theories, but at higher intensities nonlinear effects become significant. Nonlinearity leads to qualitatively new phenomena, including wave interactions and the formation of singularities (such as shock waves). This project addresses the fundamental nonlinear dynamics of surface waves and their applications, especially nondispersive surface waves whose phase speed or frequency is independent of their wavelength. As a part of the project, graduate and undergraduate students will be trained through involvement in the research.The project will carry out an asymptotic analysis of nonlinear surface waves in a variety of physical systems. Applications will be made to electromagnetic surface waves, or surface plasmons, waves on vorticity discontinuities and surface quasi-geostrophic fronts, plasma waves, and Rayleigh waves in elasticity. The resulting asymptotic equations are typically spatially nonlocal and quasilinear. These equations will be studied analytically and numerically, in particular with the aim of understanding the nonlinear focusing and possible formation and propagation of singularities in these surface waves. In addition, the project will study the diffraction and glancing Mach reflection of shock waves, together with related problems in transonic aerodynamics.