Mathematical Sciences: Linear and Nonlinear Waves in Random Media

这项建议涉及几个问题 涉及随机介质，即脉冲波的传播， 非线性方程解的奇点聚焦 薛定谔方程 特别是，首席研究员 将研究脉冲反射的正问题和逆问题， 随机介质中的波，重力反射和透射 波在不规则渠道，和奇异的解决方案， 光束非线性聚焦的薛定谔方程。 这样的问题出现在许多领域，例如地震学， 等离子体物理学和非线性光学。 许多现象都涉及波的传播， 非均匀介质 想想地震波在内部传播 地震后的地球 由于介质不均匀 或有序，我们称之为随机，科学家们使用各种 数学模型来帮助理解 性质影响宏观性质。 宏观影响 是我们可以测量的，它们通常是大多数 对我们感兴趣。 在本提案中，主要研究者将 使用分析和数值工具来研究波和光 光束在随机介质中传播时的行为。

This proposal concerns itself with several problems involving random media, namely the propagation of pulse waves and the focussing of singularities of solutions of the nonlinear Schroedinger equation. In particular, the principal investigator will study direct and inverse problems for pulse reflection of waves in random media, reflection and transmission of gravity waves in irregular channels, and singular solutions of the Schroedinger equation due to nonlinear focussing of light beams. Such problems arise in a host of areas, for example, seismology, plasma physics and nonlinear optics. Many phenomena involve the propagation of waves through inhomogeneous media. Think of seismic waves travelling inside the earth after an earthquake. Since the medium is not uniform or orderly, we call it random and scientists use a variety of mathematical models to help understand how local microscopic properties influence macroscopic ones. The macroscopic effects are the the ones we can measure and they usually are of most interest to us. In this proposal the principal investigator will use analytical and numerical tools to study how waves and light beams behave as they travel inside of random media.//