Mathematical Sciences: Nearly Integrable Nonlinear Wave Phenomena: Theory and Applications

本项目主要研究近可积非线性波 方程，即可积偏微分方程， 他们的不安。 目标之一是使用 纯数学理论、微扰理论和数值计算 实验来描述，理解和预测的行为 这些系统。 第二个目标是应用这些工具， 概念来研究非线性系统中出现的特定系统， 光学. 第三个目标是使用近似可积系统作为 提出统计性质问题的模式 非线性相互作用和非线性模式选择物理学 机制等 这里的主题是关注解决方案，首先是 精确可积解及其在 扰动 可积理论给出了显式解 以及每个N模式解的全部参数集， 可积系统内的稳定性，以及附近的解 这是由不稳定的溶液引起的。 然后数值实验 在扰动系统上测试是否近可积动力学 可以近似调制N模式可积的解决方案。 如果 从而导出了相应的摄动方程。 这些 在具体的应用项目上使用工具。 偏微分方程的可积系统通常 有解，其性质可以精确地确定。 这 与通常的非线性偏微分方程 方程 这里重要的是， 当系统受到干扰时， 可以说，一个伟大的。 处理棘手的问题 出现这样的问题 在非线性光学中（例如，在长距离的设计中）， 光纤通信网络及其设计 用于非常快的开关的光腔），在波动中 在非线性声学介质中，以及在绕薄机翼的流动中。

This project concentrates on nearly-integrable nonlinear wave equations, that is, integrable partial differential equations and their perturbations. One of the goals is to use a complement of pure mathematical theory, perturbation theory, and numerical experiments to describe, understand, and predict the behavior of these systems. A second goal is to apply these tools and concepts to study specific systems that arise in nonlinear optics. A third goal is to use nearly integrable systems as a model for formulating questions of a statistical nature about the physics of nonlinear interactions and nonlinear mode selection mechanisms. The theme here is to focus on solutions, first the exact integrable solutions and then their behavior under perturbations. The integrable theory yields explicit solutions and the full set of parameters for each N-mode solution, stability within the integrable system, and the nearby solutions that result from unstable solutions. Then numerical experiments on the perturbed systems test whether nearly-integrable dynamics can be approximated by modulated N-mode integrable solutions. If so, the corresponding perturbed equations are derived. These tools are brought to bear on specific applied projects. Integrable systems of partial differential equations often have solutions whose properties can be exactly established. This is quite unlike the usual case for nonlinear partial differential equations. What is important here is that often the properties remain when the system is perturbed. Thus one can tell a great deal about otherwise intractable problems. Such problems arise in nonlinear optics (for example, in the design of long-distance communication networks using fiber optics and the design of optical cavities for very fast switches), in the motion of waves in nonlinear acoustic media, and in flows around thin wings.