Mathematical Models of Nonlinear Wave Processes

非线性波过程的数学模型

• 批准号: 9704724
• 负责人: David Muraki
• 金额: $ 6.7万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704724&HistoricalAwards=false
• 关键词: Mathematical; Models; Nonlinear; Wave; Processes

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.

9704724 Muraki 提出的工作的主题集中在PDE模型中非线性波的产生和传播，这些模型来自当前感兴趣的物理系统： a）光在双折射液晶中的自聚焦， B）化学反应中的界面图案形成， （c）大气天气系统中的波动行为。 数学目标是量化空间结构，不稳定性和动力学之间的关系，涉及到非线性PDE系统中的波动行为。 在所有提出的项目中，模型方程涉及耦合PDE系统，其中波的行为强烈耦合到其他背景影响。 这些模型的复杂性要求发展新的方法，非线性波的联合收割机的渐近分析，现代动力系统理论和数值模拟技术相结合。 这些项目的重点是通过识别代表主要波模式的解决方案来理解非线性波，沿着它们的动力学和相互作用特性-同时在基本的物理背景下保持相关性。 自然界的变化往往是由波介导的-这是一个空间特征随时间演变的过程。 常见的例子是池塘中涟漪的向外扩散和大气湍流形式的波浪，这导致飞机经历的“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（和一般科学一样）正变得越来越量化。 解释和理解更大量的精细数据需要我们同样提高分析日益复杂的波浪生成和传播数学模型的能力。 拟议的研究旨在开发分析和计算技术，以量化光学，化学和天气系统模型方程中的波行为，并在原始科学背景下重新解释数学含义。