Mathematical Sciences: Perturbation Theory for Near-Integrable Equations and Its Application

9502142 Kovacic这项工作得到了国家科学基金教师早期职业发展奖的支持。研究的重点是近可积系统的理论。这些系统是完全可积的常微分方程和偏微分方程或积分微分方程的小扰动。将讨论两个领域：低维系统中的多脉冲同斜轨道，以及描述环腔激光光学的麦克斯韦-布洛赫积分-偏微分方程的规则和不规则动力学。本文在多脉冲同斜轨道领域的研究是作者先前在不稳定谐振系统方面工作的延续。它将在一个大的近可积系统家族中展示几种新的多脉冲轨道，从而揭示该家族中系统的复杂相空间结构。这项研究还将提供可计算的方法来验证这些复杂的同斜轨道的存在，从而在特定的例子中验证不规则动力学。这些方法在力学、流固动力学和非线性光学中的应用也被提出。在麦克斯韦-布洛赫方程领域，提出的研究包含了广泛的理论、计算和应用问题。这些问题包括寻找可积麦克斯韦-布洛赫方程的新显式解、同斜轨道和混沌动力学、有限维吸引子、激光器激发态的稳定、解的数值模拟以及光纤激光器和二极管激光器的数学描述。并与实际物理和工程应用及实验进行了比较。教育部分包括将作者的研究经验转化为以几何和动态系统为导向的方法，在二年级、三年级和研究生阶段教授微分方程课程，并为学生提供建议，并让他们参与与洛斯阿拉莫斯国家实验室的研究合作。美国国家科学基金会强烈鼓励教师作为教育工作者和研究人员的早期发展。教师早期职业发展（Career）计划是一个基金会范围内的计划，为初级教师的整体职业发展提供支持。它在一个项目中结合了对最广泛意义上的高质量研究和教育的支持，以及那些传统上在科学和工程领域代表性不足的人的充分参与。该项目加强并强调了基金会对包括研究和教育在内的全面、平衡的学术生涯发展的重要性。这个项目的研究部分包括在力学和激光光学两类物理系统中处理规则（主要是时间周期）和不规则（或混沌）行为的预期研究。这项工作将集中在接近可积的数学模型上，也就是说，模型的近似程度离显式可解只有一小步。通过忽略某些小的量，这些模型确实变得显式可解，或可积。用这种方法得到的显式解可用于近似更复杂的近可积系统的解。因此，所提出的工作将对所研究的物理系统的某些类型行为背后的机制进行数学描述，例如耦合摆振幅中的不规则节拍，以及激光的一些规则和混沌操作制度。数值计算将用于激励分析研究并证实其发现，以及将其结果扩展到数学模型，这些模型不太简化，因此不适合显式或近似解，但在物理上更准确。并与实际物理和工程应用及实验进行了比较。在这一研究过程中发现的数学技术应该具有足够的普遍性，可以应用于其他物理领域的类似问题，如非线性光纤、固体力学和流体力学。教育部分将包括将作者的研究经验纳入大二、大三、大四和研究生阶段的课堂工作，并为学生提供建议，并让他们参与与洛斯阿拉莫斯国家实验室的研究合作。

9502142 Kovacic This work is supported by a National Science Foundation Faculty Early Career Development Award. The research will focus on the theory of near- integrable systems. These systems are small perturbations of completely integrable ordinary and partial differential equations or integro-differential equations. Two areas will be addressed: multi-pulse homoclinic orbits in low-dimensional systems, and regular and irregular dynamics of the Maxwell-Bloch integro-partial differential equations that describe ring- cavity laser optics. The proposed research in the area of multi-pulse homoclinic orbits is a continuation of the author's previous work on unstable resonant systems. It will exhibit several new classes of multi-pulse orbits in a large family of near-integrable systems, and thus reveal the intricate phase-space structure of the systems in this family. This research will also provide computable methods for verifying the presence of these complicated homoclinic orbits, and therefore irregular dynamics, in specific examples. Applications of these methods in mechanics, fluid and solid dynamics, and nonlinear optics are also proposed. In the area of the Maxwell-Bloch equations, the proposed research contains a broad array of theoretical, computational, and applied questions. These questions include finding new explicit solutions of the integrable Maxwell-Bloch equations, homoclinic orbits and chaotic dynamics, finite-dimensional attractors, stabilization of the excited states of lasers, numerical simulations of solutions, and mathematical descriptions of fiber lasers and diode lasers. Comparisons with realistic physical and engineering applications and experiments are also proposed. The education component involves translating the author's research experience into a geometric and dynamical-systems oriented approach to teaching courses in differential equations on the sophomore, junior-senior, and graduate levels, and advising students and involving them in research collaborations with Los Alamos National Laboratory. The National Science Foundation strongly encourages the early development of academic faculty as both educators and researchers. The Faculty Early Career Development (CAREER) Program is a Foundation- wide program that provides for the support of junior faculty within the context of their overall career development. It combines in a single program the support of quality research and education in the broadest sense and the full participation of those traditionally underrepresented in science and engineering. This program enhances and emphasizes the importance the Foundation places on the development of full, balanced academic careers that include both research and education. The research component of this project involves intended research that addresses both regular, mainly time-periodic, and irregular, or chaotic, behavior in two classes of physical systems in mechanics and laser optics. The work will focus on mathematical models that are near-integrable, that is, models whose degree of approximation is a small step away from making them explicitly solvable. By neglecting certain small quantities, these models do become explicitly solvable, or integrable. The explicit solutions obtained in this way may be used to approximate the solutions of the more complicated near-integrable systems. The proposed work will thus develop a mathematical description of the mechanisms behind certain types of behavior of the physical systems under investigation, such as the irregular beats in the amplitudes of coupled pendula, and some of the regular and chaotic operation regimes of lasers. Numerical computations will be used to motivate the analytical investigations and confirm their findings, as well as to extend their results to mathematical models that are less simplified and thus not amenable to either explicit or approximate solution, but are more phys ically accurate. Comparisons with realistic physical and engineering applications and experiments are also proposed. The mathematical techniques discovered in the course of this investigation should be general enough to apply to similar problems in other areas of physics, such as nonlinear fiber optics, solid, and fluid mechanics. The education component will involve incorporating the author's research experience into classroom work on the sophomore, junior-senior, and graduate levels and advising students and involving them in research collaborations with Los Alamos National Laboratory.