Applied Hyperbolic Partial Differential Equations

摘要：0405899，J. Rauch，应用双曲型偏微分方程的洛根大学首席研究员将研究涉及双曲型偏微分方程的数学问题。 这些方程出现在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、电力线等。等典型的数学问题涉及显示特定的问题，由应用程序的动机，有解决方案，并研究其定性性质，例如稳定性和有效计算的方法。 第一组问题的建议涉及阻尼声波封闭声腔中的扩散介质。 有两种情况很容易理解，一是保护飞机上的驾驶员和乘客不受发动机噪音的影响，二是潜艇吸收声音，使其不会逃到海里。 也有应用于浸入流体中的机械系统的自动控制。 第二个问题是关于强非线性阻尼波动方程解的散射。 第三个问题是证明聚焦激光脉冲中的连续谱产生现象伊萨非线性电磁模型在无电离情况下的结果。 与聚焦前相比，聚焦光束中出现的频率带要宽得多，跨越了一个连续的频率（与离散的频率集相反）。 坦率地说，目前的解释并不令人信服，这是激光物理学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见解的地方。 聚焦光束被用于许多场合，包括一个令人惊讶的应用，以减少高超声速飞机的阻力。提议者将研究的微分方程的解决方案，支配波的传播行为。 研究的一个问题是，通过封闭一个区域的吸收层，声音可以被排除在该区域之外。 在这里，层中的吸收应该是扩散性质的，就像凝胶一样。 应用包括飞机、保险丝和潜艇的隔音以及浸没机械系统的控制。 要研究的第二个问题是聚焦激光器产生白色光的问题。 当高功率激光聚焦时，它们的颜色从载波频率的颜色变成接近白光。 对此有各种各样的解释，但没有一个是令人信服的。PI会努力找到更好的。 有令人信服的实验证据表明，这种现象并不需要传播介质的电离，这将有助于保持方程under研究管理和良好的基础。

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.