Mathematical Sciences: Waves in Nonlinear Dispersive and Hyperbolic Systems

首席研究员将继续他的重要研究， 非线性波动理论的三个主要领域：非线性色散 弱冲击波聚焦焦散线和灾难性 方波爆炸的不稳定性 学习方法 涉及分析和渐近技术的使用 并辅以数值分析。 这样一种综合办法 之所以需要，是因为所研究的现象发生在 现有理论的边界。 自然界中的许多现象 用线性模型充分描述;自然是固有的 “非线性”。 拟议的研究涉及波的研究 由非线性微分方程描述的运动。 一个很好的例子是“打破”的海浪，因为它 接近海滩。 为了描述这种行为， 一个非线性波，首席研究员将使用渐近 分析与数值分析相结合，以获得定性 和定量信息。

The principal investigator will continue his important studies of three major areas of nonlinear wave theory: nonlinear dispersive waves, focussing of weak shock waves at caustics and catastrophic instabilities in square-wave detonations. The methods of study involve the use of analytic and asymptotic techniques supplemented with numerical analysis. Such a combined approach is called for because the phenomena under study occur at the boundaries of existing theories. Many phenomena in nature cannot be described adequately with linear models; nature is inherently "nonlinear". The proposed research involves the study of wave motions that are described by nonlinear differential equations. A good example is the "breaking" of an ocean wave as it approaches the beach. In order to describe the behavior of such a nonlinear wave the principal investigator will use asymptotic analysis in concert with numerical analysis to obtain qualitative and quantitative information.