Applied Hyperbolic Partial Differential Equations

摘要：0405899，J·劳赫，密歇根大学应用双曲型偏微分方程首席研究员将研究涉及双曲型偏微分方程式的数学问题。这些方程出现在对以有限速度传播的波的描述中，例如声学、电磁学、流体和空气动力学、等离子体、激光、电报、无线电、电视、手机、冲击波、音乐、电力线……典型的数学问题包括证明特定的问题在应用的推动下有解，并研究其定性性质，例如可例性和有效计算的方法。建议中的第一组问题涉及通过将声腔封闭在扩散介质中来抑制声波。需要了解的两种情况是保护飞机上的飞行员和乘客免受发动机噪音的影响，以及潜艇吸收声音，使其不会逃往大海。也有应用于浸入流体中的机械系统的自动控制。第二个问题涉及一个强非线性阻尼型波动方程的解的散射。第三个问题是证明聚焦激光脉冲中的连续谱产生现象是没有电离的非线性电磁模型的结果。自聚焦激光光束具有连续谱产生。与聚焦前相比，聚焦光束中存在的频带频率要宽得多，并且跨越了连续的频率(与离散的集合形成对比)。坦率地说，目前的解释并不十分令人信服。这是激光物理界长期以来认识到的一个基本问题。这是一个数学可以增加重要洞察力的地方。聚焦波束被用于许多情况，包括在高超声速飞行器中用于减阻的紧急应用。作者将研究控制波传播的微分方程解的行为。研究的一个问题是，通过将一个区域封闭为一个吸声层，可以在多大程度上将声音排除在该区域之外。在这里，层中的吸收应该是扩散性质的，就像人们对凝胶的吸收一样。应用包括飞机机身和潜艇的声波隔音，以及对浸入式机械系统的控制。第二个要研究的问题是聚焦激光中产生白光的问题。当高功率激光聚焦时，它们的颜色从载波频率的颜色变为近乎白光。对此有各种各样的解释，但没有一个很有说服力。私家侦探会试着找一个更好的。有令人信服的实验证据表明，这种现象不需要电离传播介质，这将有助于保持所研究的方程是可管理的和有充分依据的。

Abstract: 0405899, J. Rauch, University of MichiganApplied Hyperbolic Partial Differential Equations The principal investigator will investigate mathematical problemsinvolving hyperbolic partial differential equations. These equationsappear in the description of waves which propagate with finite speed,for example acoustics, electromagnetism, fluid and aero dynamics,plasmas, lasers, telegraphy, radio, television, cell phones, shockwaves, music, power lines, .... etc. Typical mathematical problemsinvolve showing that specific problems, motivated by applications,have solutions and studying their qualitative properties, for examplestability and methods for effective computation. A first set ofproblems in the proposal concerns the damping of acoustic waves byenclosing the acoustic cavity in a diffusive medium. Two situationseasy to understand are the protection of pilots and passengers inairplanes from the noise of the engines and the absorption of sound ina submarine so that it will not escape to the sea. There are alsoapplications to the automatic control of mechanical systems immersedin fluids. A second problem concerns the scattering of solutions of astrongly nonlinearly damped wave equation. A third problem is to showthat the phenomenon of continuum generation in focused laser pulses isa consequence of nonlinear electromagnetic models without ionization.Self focused laser beams exhibit continuum generation. The band offrequencies present in the focused beam is much broader and spans acontinuum of frequencies (in contrast to a discrete set) than beforethe focusing. Current explanations are frankly not very convincing.This is a fundamental problem long recognized in the laser physicscommunity. It is a place where mathematics may add importantinsights. Focused beams are used in many situations including asurprising application to reduce resistance in hypersonic aircraft.The proposer will study the behavior of solutions of the differentialequations that govern the propagation of waves. One probleminvestigated is how well sound can be excluded from a region byenclosing the region is an absorbing layer. Here the absorption inthe layer is supposed to be of a diffusive nature, like one would havefor a gel. Applications include sonic insulation of airplanefuselages and submarines and the control of immersed mechanicalsystems. A second problem to be studied is that of white lightgeneration in focused lasers. When high power lasers focus, theircolor changes from the color of the carrier frequency to nearly whitelight. There are a variety of explanations of this, none veryconvincing. The PI will try to find a better one. There is convincingexperimental evidence that the phenomenon does not require ionizationof the propagation medium and this will help to keep the equationsunder study manageable and well founded.