Mathematical Sciences: Linear and Nonlinear Waves in Random Media

数学科学：随机介质中的线性波和非线性波

• 批准号: 9003227
• 负责人: George Papanicolaou
• 金额: $ 16.4万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-01 至 1993-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9003227&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Linear; Nonlinear; Waves

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.//

该提案涉及涉及随机培养基的几个问题，即脉搏波的传播以及非线性施罗丁格方程解决方案溶液的奇异性的焦点。 特别是，主要研究者将研究随机培养基中波的脉冲反射，不规则通道中重力波的反射和传播的直接和反问题，以及由于光束非线性焦点而引起的Schroedinger方程的奇异溶液。 此类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器件。 许多现象涉及通过不均匀介质传播波浪的传播。 想想地震后地震浪潮在地球内传播。 由于该介质不是统一或有序的，因此我们称其为随机，科学家使用各种数学模型来帮助了解局部微观特性如何影响宏观的微观特性。 宏观效应是我们可以衡量的效果，它们通常是我们最感兴趣的。 在此提案中，主要研究人员将使用分析和数值工具来研究波浪和光束在随机介质内部传播时的行为。////