Dynamics of Nonlinear Waves in Optical Systems

本文主要研究非线性波及其在非线性光学系统中的应用。利用现代微扰方法、严格的分析和科学计算的结合，将探索和发展各种各样的光学模型。结合大量的建模工作，数学目标将是进一步发展和扩展用于量化和理解在光孤子传输、锁模光纤激光器和光纤网络设备和应用研究中出现的非线性、色散偏微分方程波动力学的方法。特别是，将采用各种分析方法，目的是将哈密顿系统和可积系统中的控制方程简化为更容易处理的偏微分方程组，这些偏微分方程组仍然保留完整控制方程的基本特征。由此产生的动力学将被完全分类，并特别注意系统中可能出现周期性或混沌动力学的参数区域。尽可能地进行简化模型之间的比较、控制方程的数值模拟和实验。这项研究的目标是识别、分类和模拟光学领域当前感兴趣的物理系统。将特别关注贝尔实验室(朗讯技术)、泰科潜艇系统国际公司、3M公司、休斯研究实验室和霍尼韦尔公司等工业合作伙伴最感兴趣的应用。随着光纤技术的重要性增加，光纤应用已从单纯的高数据速率通信(孤子传输)扩展到廉价的光纤激光光源、局域网设计和应用，以及全光网络设备和组件。该提案涵盖了目前在这些领域的研究工作的所有领域。特别是，与工业伙伴的密切工作关系确保了在最重要的应用领域进行及时的研究。有了这些工业合作伙伴提供的现成的实验研究，可以构建和开发精确的数学模型，以捕捉每个光学系统的基本方面：因此，数学模型有可能对现实和物理上可实现的系统的设计和实施产生重大影响。

This research focuses on the study of nonlinear waves and their applications in nonlinear optical systems. Using a combination of modern perturbation methods, rigorous analysis, and scientific computation, a wide variety of optical models will be explored and developed. In conjunction with significant modeling efforts, the mathematical objectives will be to further develop and extend the methods utilized for quantifying and understanding the wave dynamics of nonlinear, dispersive partial differential equationswhich arise in the study of optical soliton transmission, modelocked fiber lasers, and optical fiber network devices and applications. In particular, a variety of analytic methods will be pursued with the goalof reducing the governing equations in Hamiltonian and integrable systems to more readily handled systems of partial differential equation that still retain the fundamental characteristics of the full governing equations. The resulting dynamics will be completely classified with particular attention given to parameter regimes for which periodic or chaotic dynamics may arise in the system. Comparison between the reduced models, numerical simulations of the governing equations, and experiments will be performed whenever possible. The goal of this research is to identify, classify, and model physical systems of current interest in the optics community. Particular interest will be given to applications of greatest interest to industrial partners at Bell Laboratories (Lucent Technologies), Tyco Submarine Systems International, the 3M corporation, Hughes Research Labs, and Honeywell Inc. With the increased importance of optical fiber technology, fiber optic applications have broadened from mere high data-rate communications (soliton transmission) to inexpensive optical fiber laser sources, local area network design and applications, and all-optical network devices and components. The proposal covers all areas of current research efforts in these areas. In particular, the close working relations with industrial partners ensures timely research in the most important areas of application. And with the readily available experimental studies provided by these industrial partners, accurate mathematical models can be constructed and developed to capture the fundamental aspectsof each optical system: thus providing the potential for the mathematical models to have a significant impact on design and implementation of realistic and physically realizable systems.