Nonlinear wave propagation

NSF奖摘要-DMS-0072343数学科学：非线性波传播这个项目研究了非线性波传播中的一些问题。第一个问题是弱冲击波的反射，在这方面，理论和实验之间存在着长期的差异。这一问题与跨音速空气动力学有着密切的关系。第二个问题是研究广义相对论爱因斯坦场方程中引力波的非线性效应，这是一个具有基础科学意义的问题。第三个问题是弹性力学和磁流体力学中双曲面波的非局部、非线性方程的研究。例如，这种波出现在信号处理中使用的表面声波设备中。第四个问题是研究不可压缩流体中高频涡度波与平均流的相互作用。这项研究将描述涡度不稳定性的非线性发展，并与湍流的闭合问题有关。第五个问题是反应扩散方程的双稳态振荡系统中前锋的传播，该方程提供了胰腺中产生胰岛素的β细胞集体行为的简化模型。第六个问题是表面活性剂在扩散液滴中沉积的研究。这一问题在使用液滴沉积表面膜方面具有工业应用。声波、弹性波和引力波等波是许多物理和生物系统的重要特征。大振幅波的行为可能是非线性的，这导致了在线性波中看不到的新效应：例如，以接近或高于音速的速度飞行的飞机产生冲击波。对描述非线性波的方程进行详细分析通常是非常困难的。这项研究的目的是在各种应用的背景下增加对非线性波的理解，这些应用涉及定性上有趣而鲜为人知的现象，以及与当前科学和技术感兴趣的问题有关的问题。

NSF Award Abstract - DMS-0072343Mathematical Sciences: Nonlinear Wave PropagationAbstract0072343 HunterThis project studies a number of problems in nonlinear wave propagation. The first problem is the reflection of weak shock waves, where there have been longstanding discrepancies between theory and experiment. There are close relationships between this problem and transonic aerodynamics. The second problem is the study of nonlinear effects on gravitational waves in the Einstein field equations of general relativity, an issue of fundamental scientific significance. The third problem is the study of nonlocal, nonlinear equations for hyperbolic surface waves in elasticity and magnetohydrodynamics. Such waves arise, for example, in surface acoustic wave devices used in signal processing. The fourth problem is the study of the interaction of high frequency vorticity waves and mean flows in incompressible fluids. This research will describe the nonlinear development of vorticity instabilities, and is relevant to the closure problem for turbulent flows. The fifth problem is the propagation of fronts in a bistable oscillatory system of reaction-diffusion equation that provides a simplified model of the collective behavior of beta-cells in the pancreas, which produce insulin. The sixth problem is the study of surfactant deposition by a spreading liquid drop. This problem has industrial applications in the use of droplets for the deposition of surface films. Waves, such as sound waves, elastic waves, and gravitational waves, are an important feature of many physical and biological systems. Large amplitude waves may behave nonlinearly, and this leads to new effects not seen in linear waves: for example, the generation of shock waves by an aircraft traveling at speeds close to or above the speed of sound. A detailed analysis of the equations that describe nonlinear waves is often very difficult. The aim of this research is an increased understanding of nonlinear waves in the context of a variety of applications that involve qualitatively interesting and poorly understood phenomena, and that are related to problems of current scientific and technological interest.