Mathematical Sciences: Nonlinear Dispersive Waves

在非线性色散波的理论与应用、混沌非线性波的具体领域、大气与海洋相互作用中产生的波、液晶非线性光学等方面继续进行研究。他在混沌非线性波的数学理论方面的工作将集中在偏微分方程的同斜轨道的持久性和后果上。在该项目中，他开发了偏微分方程的几何奇异摄动方法，包括不变流形振动和范式。然后，他将这些来自动力系统理论的方法与Melnikov分析相结合，证明了同斜轨道的持久性。然后用数值方法研究了这些轨道对非线性波动系统的影响。这种非线性色散波数学理论中的现代方法被应用于大气、海洋物理和非线性光学等领域。大气-海洋相互作用和非线性光学中的激光传播是科学中出现非线性色散波的两个最自然的领域。提出者将发展这些波的现代数学理论，并将其应用于这些科学领域。在这些非线性波动系统的时间演化中出现的混沌和不规则行为，刚刚开始在数学上得到理解。拟研究这种非线性和不规则行为对海面温度振荡、风暴路径预测和演变以及激光穿过液晶材料传播的影响。这项科学研究的方法包括数学理论、形式摄动理论和科学计算。＊＊＊

DMS-9600128 McLaughlin The proposer continues his research in the theory and application of nonlinear dispersive waves, in the specific areas of chaotic nonlinear waves, waves which occur in the interaction of the atmosphere with the ocean, and topics in liquid crystal nonlinear optics. His work on the mathematical theory of chaotic nonlinear waves will focus upon the persistence and consequences of homoclinic orbits for partial differ- ential equations. In the project he develops geometric singular perturbation methods for partial differential equations, including including invariant manifold fibrations and normal forms. He then combines these methods from dynamical systems theory with Melnikov analysis to prove the persistence of homoclinic orbits. Consequences of these orbits for nonlinear wave systems are then investigated numerically. Such modern methods in the mathematical theory of nonlinear dispersive waves are applied to topics in atmosphere ocean physics and to nonlinear optics. %%% Atmosphere ocean interactions and laser propagation in nonlinear optics are two of the most natural areas in science for the occurence of nonlinear dispersive waves. The proposer will develop modern mathe- matical theory of these waves, and apply it to these areas of science. Chaotic and irregular behavior, which occurs in the temporal evolution of these nonlinear wave systems, is just becoming understood mathematically. The proposer will study the effects of such nonlinear and irregular behavior on oscillations of the sea surface temperature, on the prediction and evolution of storm tracks, and on the propagation of laser light through liquid crystal materials. Methods for this scientific study include mathematical theory, formal perturbation theory, and scientific computation. ***