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。它集中于由非线性双曲偏微分方程组和相关方程模拟的波，沿着边界或界面传播(例如涡量、涡片、物质边界和激波中的不连续)。这些表面波通常表现出一种复杂的非局部、非线性行为，这种行为还没有被很好地理解。主要研究人员将推导和研究描述这些波在一系列物理应用中的简化的渐近方程。所得到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来表示，这导致了与多线性调和分析的联系。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。面波是沿着边界或界面传播的波。因为它们是沿着界面引导的，所以它们的衰减速度比体波慢，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破坏性最强。由于表面波可直接用于检测和操纵，因此被广泛应用于技术应用中，如手机中的超声波表面声波器件或纳米光子表面等离子体器件。小幅度波可以用线性方程很好地描述，但非线性效应在较大幅度时变得重要，并导致新的定性现象，如奇点的形成(例如，可压缩流体中的冲击波)。非线性使得对这些问题的数学分析非常具有挑战性。面波的另一个特征是，非线性的影响可能是非局部的，因为表面上某一点发生的事情会通过主体介质影响到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非线性、非局部表面波的基本性质。这些结果将在流体动力学方面有潜在的应用，包括跨音速流动、弹性、磁流体动力学、地球物理和凝聚态物理。

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.