Nonlinear Wave Propagation

摘要：提案0309648，Hunter，加州大学戴维斯分校标题：非线性波传播这个项目解决了双曲守恒定律理论中基本的和鲜为人知的问题，例如那些模拟可压缩流体力学的问题。它还解决了不可压缩流体力学和广义相对论的问题，并将结果应用于各种重要的物理问题。许多提出的工作的一个共同主题是非线性如何导致各种奇点的形成和传播，例如可压缩流体中的激波，或广义相对论中的时空奇点。提出的具体研究课题有：激波反射；激波在随机介质中的传播；弹性与磁流体力学中的非线性双曲表面波不可压缩流体力学中的普朗特边界层方程和涡量波、平均流动相互作用；以及广义相对论对引力波非线性的影响。波（如声波、地震波、光波、量子力学波或引力波）是许多物理系统的中心特征，从原子尺度到宇宙尺度。在较低强度下，波重叠，它们可以用线性理论来描述，但在较高强度下，非线性效应变得重要，导致定性的新现象（如激波和孤子）。非线性波比线性波更难用数学方法分析，而且往往具有重要的实际意义。例如，在接近音速的飞机周围形成的冲击波。该项目的目的之一是研究基本的，但尚未完全理解的冲击波现象，例如它们从墙壁上的反射。在其他课题中，该项目还包括非线性表面波的研究，例如由地震产生并用于各种超声波设备的瑞利波，以及引力波中时空奇点的形成和传播的研究。

Abstract, Proposal 0309648, Hunter, University of California-DavisTitle: Nonlinear wave propagationThis project addresses fundamental and poorly understood issues in thetheory of hyperbolic conservation laws,such as the ones that modelcompressible fluid mechanics. It also addresses problems inincompressible fluid mechanics and general relativity, and applies theresults to a variety of important physical problems. A common theme ofmuch of the proposed work is how nonlinearity leads to the formationand propagation of various kinds of singularities, such as shock wavesin compressible fluids, or space-time singularities in generalrelativity. Specific topics proposed for study are the following:shock wave reflection; shock wave propagation in random media;nonlinear hyperbolic surface waves in elasticity andmagnetohydrodynamics; the Prandtl boundary layer equations andvorticity wave, mean flow interactions in incompressible fluidmechanics; and the effect of nonlinearity on gravitational waves ingeneral relativity.Waves (such as sound waves, seismic waves, light waves, quantummechanical waves,or gravitational waves) are a central feature of manyphysical systems, from the atomic to the cosmological scales. At lowerintensities, waves superpose and they may be described by lineartheories, but at higher intensities nonlinear effects becomessignificant, leading to qualitatively new phenomena (like shock wavesand solitons). Nonlinear waves are much more difficult to analysemathematically than linear waves, and are often of great practicalimportance. For example, shock waves form around aircraft that travelclose to the speed of sound. One aim of this project is to studybasic, but not fully understood, shock wave phenomena, such as theirreflection from walls. Among other topics, the project also includes astudy of nonlinear surface waves, such as the Rayleigh waves which aregenerated by earthquakes and are used in a variety of ultrasonicdevices, and a study of the formation and propagation of space-timesingularities in gravitational waves.