Mathematical Models of Nonlinear Wave Processes

9704724 Muraki拟议工作的主题集中在PDE模型中非线性波的产生和传播，这些波来自当前感兴趣的物理系统：a)向列相液晶中光的自聚焦，b)化学反应中界面图案的形成，c)大气天气系统中的波行为。数学目标是量化空间结构、不稳定性和动力学之间的关系，因为它们与非线性偏微分方程系统中的波动行为有关。在所有提出的项目中，模型方程涉及耦合的PDE系统，其中波的行为与其他背景影响强耦合。这些模型的复杂性要求发展结合渐近分析、现代动力系统理论和数值模拟技术的非线性波的新方法。这些项目侧重于通过确定代表主波模式的解决方案以及它们的动力学和相互作用属性来了解非线性波，同时保持与基本物理环境的相关性。自然界的变化通常由波动来调节--在这个过程中，空间特征会随着时间的推移而演变。常见的例子是池塘中的涟漪向外扩散，以及以大气湍流的形式出现的波浪，这导致了飞机经历的“颠簸”飞行。由于更高精度的仪器和更大规模的计算，波浪科学(就像一般的科学一样)正变得越来越定量。为了解释和理解更多的精细数据，我们需要同样增强分析日益复杂的波浪产生和传播的数学模型的能力。这项拟议的研究旨在开发分析和计算技术，以量化光学、化学和天气系统的模型方程中的波动行为，并在原始科学背景下重新解释数学含义。

9704724 Muraki The theme of proposed work centers on the generation and propagation of nonlinear waves in PDE models which derive from physical systems of current interest: a) self-focussing of light in nematic liquid crystals, b) interfacial pattern formation in chemical reactions, c) wave behaviors in atmospheric weather systems. The mathematical objective is to quantify relationships between spatial structure, instability, and dynamics as pertains to wave behavior in nonlinear PDE systems. In all of the proposed projects, the model equations involve coupled PDE systems where the wave behavior is strongly coupled to other background influences. The complexity of these models requires the development of new methods for nonlinear waves which combine the techniques of asymptotic analysis, modern dynamical systems theory, and numerical simulation. These projects focus on the understanding of nonlinear waves through the identification of solutions representing the primary wave modes, along with their dynamics and interaction properties -- while remaining relevant within the underlying physical context. Change in nature is often mediated by waves -- a process in which spatial features evolve over time. Familiar examples are outward spreading of ripples in a pond and waves in the form of atmospheric turbulence, which result in the "bumpy" flights experienced by aircraft. The science of waves (as with science in general) is becoming increasingly quantitative, due to higher-precision instrumentation and larger-scale computing. The interpretation and understanding of greater amounts of refined data demands an equal enhancement of our ability to analyze increasingly sophisticated mathematical models of wave generation and propagation. The proposed research seeks to develop analytical and computational techniques to quantify wave behavior in model equations for optical, chemical, and weather systems and to re-interpret the mathem atical implications within the original scientific context.