Quasi-linear hyperbolic and surface waves

该项目涉及各种物理系统中非线性波传播的数学建模和分析。它侧重于非色散波，特别是在界面上传播的表面波，如涡度不连续面、涡片、物质边界、水波和激波。在该建议中考虑的许多波动在线性化极限中具有恒定的非零频率。这些波形成了相对较少研究的一类非色散波，提出的研究旨在发展对其非线性动力学的理解，这与色散波或非色散双曲波的性质不同。提出的研究将推导和研究这些波的渐近描述，并将开发准线性波动方程的范式变换。哈密顿动力学为所研究的大多数非线性波动提供了统一的框架。对于小振幅波，这种描述更容易以频谱形式进行，由于其相互作用的空间非局域性，这尤其适用于所提出研究中考虑的表面波。理解由此产生的非线性动力学的频谱和空间描述之间的关系是一个基本问题，也是与许多其他问题相关的问题。提出的研究的另一个主题是研究激波的掠掠马赫反射。激波反射是双曲守恒律中最重要的多维问题之一，导致了非常有趣和复杂的现象。这些结果对跨声速空气动力学的相关问题也有一定的启示。表面波是沿边界或界面传播的波。最熟悉的例子是水体（如海洋）表面的水波。另一种表面波由固体界面上的瑞利波组成。这些波是由地震产生的，它们也被用于技术应用，比如手机上的超声波表面声波设备。另一个例子是金属和绝缘体之间的界面上的电磁表面波或表面等离子体，它在光子学中得到了应用。小振幅的波可以用线性方程很好地描述，但在较大的振幅下，非线性效应变得很重要。这些效应导致了性质上的新现象，如破波、激波或其他奇点的形成，以及非线性波相互作用产生的新波。非线性，以及随波移动的自由表面的可能性，使得这些问题的数学分析非常具有挑战性。表面波的另一个特征是非线性的影响可能是非局部的，因为在表面上一点发生的事情可以通过体介质影响表面上其他地方发生的事情。首席研究员计划在各种物理问题的背景下研究这种表面波的基本定性性质。研究结果将在流体动力学、跨声速流动、弹性、磁流体动力学、地球物理学和凝聚态物理等领域具有潜在的应用前景。

This project addresses the mathematical modeling and analysis of nonlinear wave propagation in a variety of physical systems. It focuses on nondispersive waves, especially surface waves that propagate on interfaces such as discontinuities in vorticity, vortex sheets, material boundaries, water waves, and shock waves. Many of the wave motions considered in the proposal have constant, nonzero frequency in the linearized limit. These waves form a comparatively little studied class of nondispersive waves, and the proposed research aims to develop an understanding of their nonlinear dynamics, which is qualitatively different from that of dispersive waves or nondispersive hyperbolic waves. The proposed research will derive and study asymptotic descriptions of these waves and will also develop normal form transformations for quasi-linear wave equations. Hamiltonian dynamics provides unifying framework for most of the nonlinear wave motions to be studied in the proposed research. For small-amplitude waves, this description is more easily carried out in spectral form, which is particularly appropriate for the surface waves considered in the proposed research because of the spatial nonlocality of their interactions. The issue of understanding the relationship between the spectral and spatial descriptions of the resulting nonlinear dynamics is a fundamental one and one that is relevant to many other problems. A further topic of the proposed research is a study of the glancing Mach reflection of shock waves. Shock reflection is one of the most important multi-dimensional problems for hyperbolic conservation laws, leading to remarkably interesting and complex phenomena.These results should also shed light on related problems in transonic aerodynamics.Surface waves are waves that propagate along a boundary or interface. The most familiar example consists of the water waves on the surface of a body of water, like an ocean. Another type of surface wave consists of the Rayleigh waves on a solid interface. These waves are generated by earthquakes, and they are also used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones. A further example consists of the electromagnetic surface waves, or surface plasmons, on the interface between a metal and an insulator, which find applications in photonics. Small-amplitude waves are well-described by linear equations, but at larger amplitudes nonlinear effects become important. These effects lead to qualitatively new phenomena such as wave-breaking, the formation of shock waves or other singularities, and the generation of new waves by nonlinear wave-interactions. Nonlinearity, and the possibility of a free surface that moves with the wave, makes the mathematical analysis of these problems very challenging. An additional feature of surface waves is that the effects of nonlinearity may be nonlocal because what happens at one point on the surface can influence what happens elsewhere on the surface through the bulk medium. The principal investigator plans to study the fundamental qualitative properties of such surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.