Mathematical Sciences: Coherence and Chaos in PDE's and Nonlinear Wave Propagation

本项目涉及研究具有波动现象的非线性动力系统。它有两个主要目标：(1)了解偏微分方程中的混沌行为在多大程度上是有限维的；(2)研究各种传播现象。在目标(1)下，研究将集中于两类由偏微分方程描述的系统：由强迫阻尼非线性薛定谔方程和由正弦戈登方程描述的近可积系统。它们是强迫阻尼摆的偏微分方程的自然延伸，该问题被用作研究常微分方程混沌的原型。这些方程有一个自然选择的基，有了它，非线性功率谱的概念可以第一次被明确地描述。这些方程的近可积性提供了足够的分析控制，以产生关于定性行为的精确、详细的理论信息。非可积系统包括有限几何流动、湍流尾迹和边界层。这里的问题是找到流场的最优分解，其中主导动力学的模态子空间的维数最小。在目标(2)中，将研究非线性波在各种情况下的传播，试图回答诸如在随机介质中传播的非线性波的Anderson局域性的存在性、非线性光学的理论基础以及包络方程奇点的数学理论等问题。这项研究是我国近十年来在动力系统非线性分析方面所作的更大努力的一部分。这样的研究对于深入理解技术中遇到的现有动力系统中的许多非线性现象是很重要的。

This project involves research on nonlinear dynamical systems exhibiting wave phenomena. It has two main goals: (1) to understand to what extent the chaotic behavior in partial differential equations is finite dimensional; (2) to study a variety of propagation phenomena. Under goal (1) the investigations will focus on two kinds of systems described by partial differential equations: nearly integrable systems described by forced damped nonlinear Schrodinger equation and by sine-Gordon equation. They are natural extensions into PDE's of forced damped pendulum, a problem used as a prototype for studying chaos in ordinary differential equations. These equations have a natural choice of a basis, with which the notion of a nonlinear power spectrum can be described explicitly for the first time. The near integrability of these equations provides sufficient analytical control to yield precise, detailed theoretical information about the qualitative behavior. The far from integrable systems include flows in finite geometries, turbulent wakes and boundary layers. Here the issue is to find an optimal decomposition of the flow field for which the dimension of the subspace of modes which dominate the dynamics is minimal. Under goal (2), the propagation of nonlinear waves will be studied in a variety of contexts, trying to answer questions such as the existence of Anderson localization in nonlinear waves propagating in a random medium, theoretical foundations of nonlinear optics, and mathematical theory of singularities of envelope equations. This research is part of a larger effort in the nonlinear analysis of dynamical systems which has been developed in this country in the last ten years. Studies like this are important for a deep understanding of a number of nonlinear phenomena in the existing dynamical systems encountered in technology.