Nonlinear hyperbolic waves and interfaces

该项目致力于连续介质力学中非线性、非色散波传播的数学建模和分析。它着重于由非线性双曲偏微分方程和相关方程模拟的波，这些波沿沿着边界或界面传播（如涡量、涡面、材料边界和冲击波中的不连续性）。这些表面波往往表现出复杂的非局部，非线性的行为，这是没有很好地理解。主要研究者将推导和研究描述这些波在一系列物理应用中的简化渐近方程。由此产生的非局部拟线性方程的一个典型特征是它们是哈密顿的，并且它们可以用谱和空间形式表示，从而与多线性调和分析相联系。关于这些方程的基本问题包括光滑解的寿命，奇点的形成和物理解释，以及弱解的整体存在性。表面波是沿沿着边界或界面传播的波。由于它们是沿着一个界面沿着传播的，所以它们比体波衰减得慢，这就解释了为什么地震产生的地表地震波在远离震源的地方是最具破坏性的。表面波广泛用于技术应用中，例如手机中的超声表面声波装置或纳米光子表面等离子体激元装置，因为它们可直接用于检测和操纵。小振幅波可以用线性方程很好地描述，但非线性效应在较大振幅时变得重要，并导致定性的新现象，如奇点的形成（例如，可压缩流体中的冲击波）。非线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效应可能是非局部的，因为在表面上的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各样的物理问题的背景下研究这种非线性，非局部表面波的基本定性性质。这些结果在流体力学，包括跨音速流动、弹性力学、磁流体力学、微物理学和凝聚态物理学中具有潜在的应用。

The project addresses the mathematical modeling and analysis of nonlinear, nondispersive wave propagation in continuum mechanics. It focuses on waves modeled by nonlinear hyperbolic PDEs, and related equations, that propagate along boundaries or interfaces (such as discontinuities in vorticity, vortex sheets, material boundaries, and shock waves). These surface waves often display a complex nonlocal, nonlinear behavior which is not well-understood. The principal investigator will derive and study reduced asymptotic equations that describe these waves in a range of physical applications. A typical feature of the resulting nonlocal quasilinear equations is that they are Hamiltonian, and they may be expressed in both spectral and spatial forms, leading to connections with multilinear harmonic analysis. Fundamental questions concerning these equations include the life-span of smooth solutions, the formation and physical interpretation of singularities, and the global existence of weak solutions.Surface waves are waves that propagate along a boundary or interface. Since they are guided along an interface, they decay more slowly than bulk waves, which explains why the surface seismic waves generated by an earthquake are the most destructive far from their source. Surface waves are widely used in technological applications, such as ultrasonic surface acoustic wave devices in cell phones or nanophotonic surface plasmon devices, because they are directly accessible to detection and manipulation. Small-amplitude waves are well-described by linear equations, but nonlinear effects become important at larger amplitudes and lead to qualitatively new phenomena such as the formation of singularities (for example, shock waves in a compressible fluid). Nonlinearity 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 nonlinear, nonlocal surface waves in the context of a wide variety of physical problems. The results will have potential applications in fluid dynamics, including transonic flow, elasticity, magnetohydrodynamics, geophysics, and condensed matter physics.