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

本项目主要研究几乎可积的非线性波动方程，即可积偏微分方程组及其扰动。其中一个目标是使用纯数学理论、微扰理论和数值实验的补充来描述、理解和预测这些系统的行为。第二个目标是应用这些工具和概念来研究出现在非线性光学中的特定系统。第三个目标是使用几乎可积的系统作为一个模型，用来阐述关于非线性相互作用的物理和非线性模式选择机制的统计性质的问题。这里的主题是关注解，首先是精确的可积解，然后是它们在扰动下的行为。可积理论给出了显式解和每个N模解的完整参数集，可积系统内的稳定性，以及由不稳定解产生的邻近解。然后，对摄动系统进行数值实验，检验近可积动力学能否用调制的N-模可积解来逼近。如果是这样，则导出相应的摄动方程。这些工具被用于具体的应用项目。可积偏微分方程组的解的性质通常是可以精确确定的。这与通常的非线性偏微分方程解的情况完全不同。这里重要的是，当系统受到干扰时，这些属性通常会保留下来。因此，人们可以说出许多原本难以解决的问题。这些问题出现在非线性光学中(例如，在使用光纤的长距离通信网络的设计和用于快速开关的光腔的设计中)，在非线性声学介质中的波的运动，以及在薄翼周围的流动中。

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.