Mathematical Sciences: Nonlinear Dispersive Waves

9302988 McLaughlin研究者使用数学理论、形式近似方法和科学计算相结合的方法，继续他在非线性波理论方面的研究。此外，非线性光学的一些项目将与普林斯顿大学物理系的成员合作完成，他们将为研究提供直接的实验组成部分。McLaughlin将研究非线性波的三个一般领域——(i)时间混沌波，（ii）非线性光学，（iii）空间和时间混沌波。在时间混沌非线性波的项目中，来自动力系统理论的数学方法将适用于波的偏微分设置。时空混沌波项目将发展波的偏微分方程的随机和动态数学方法。非线性光学的项目将集中于激光与液晶的相互作用。由于液晶介质是非常非线性的，它们为研究“光-物质”相互作用提供了极好的材料。自然界中的所有非线性波，从激光物理中的光波到海洋表面的水波，都可以用偏微分方程（即非线性波动方程）进行数学建模。虽然线性波动方程已经被很好地理解了一段时间，但科学家和数学家对非线性波动方程解的行为知之甚少，直到最近科学计算能力的爆发。虽然科学计算可以在非线性波动方程的解中提出和发现新的和意想不到的行为，但人们永远无法确定数值实验的结果。通常需要对进一步理想化的模型进行数学研究——不仅要确保数值实验的有效性，而且要在模型中建立定性行为。有了模型非线性偏微分方程的确定结果，物理实验就可以用来验证波动方程是对自然界中真实波动的描述。本项目与“光-液晶”相互作用的非线性光学实验一起，在非线性波的数学研究中解决了这些相关领域（科学计算、数学研究和物理实验）。＊＊＊

9302988 McLaughlin The investigator continues his research in the theory of nonlinear waves, using a combination of mathematical theory, formal approximation methods, and scientific computation. In addition, some of the projects in nonlinear optics will be done in collaboration with members of the Department of Physics at Princeton University who will provide a direct experimental component to the research. McLaughlin will work in three general areas of nonlinear waves -- (i) temporally chaotic waves, (ii) nonlinear optics, and (iii) spatially and temporally chaotic waves. In the project on temporally chaotic nonlinear waves, mathematical methods from dynamical systems theory will be adapted to the partial differential setting of waves. The project on spatially and temporally chaotic waves will develop both stochastic and dynamical mathematical methods for the partial differential equations of waves. The projects in nonlinear optics will focus upon laser light interacting with liquid crystals. Because liquid crystal media are extremely nonlinear, they provide excellent materials in which to study ``light-matter''interactions. All nonlinear waves in nature, from the light waves of laser physics to water waves on the surface of the ocean, are modeled mathematically by partial differential equations known as nonlinear wave equations. While linear wave equations have been well understood for some time, scientists and mathematicians knew little about the behavior of solutions of nonlinear wave equations until the recent explosion in scientific computing power. While scientific computation can suggest and discover new and unexpected behavior in solutions of nonlinear wave equations, one can never be certain of the outcome of numerical experiments. Mathematical studies, often of further idealized models, are required -- not only to ensure the validity of the numerical experiments, but also to establish qualitative behavior in the model. With results certain for the model nonlinear partial differential equations, physical experiments can then be used to validate the wave equations as a description of real waves in nature. This project, taken together with the experiments in nonlinear optics on the ``light-liquid crystal'' interaction, addresses each of these related areas (scientific computation, mathematical studies, and physical experiments) in the mathematical studies of nonlinear waves. ***