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

小行星9502142 这项工作得到了美国国家科学基金会的支持， 职业发展奖。 研究的重点是近可积系统的理论.这些系统都是小扰动的， 可积常微分和偏微分方程或积分微分方程 方程两个领域将得到解决：多脉冲同宿轨道， 低维系统，以及规则和不规则的动力学 描述环形腔激光光学的麦克斯韦-布洛赫积分-偏微分方程。在多脉冲领域的拟议研究 同宿轨道是作者以前工作的继续， 不稳定的共振系统它将展示几种新的多脉冲类型 轨道在一个大家庭的近可积系统，从而揭示了 复杂的相空间结构。 本研究 还将提供可计算的方法来验证这些 复杂的同宿轨道，因此不规则的动力学，在特定的 例子. 这些方法在力学、流体和固体中的应用 动力学和非线性光学也被提出。的区域中 麦克斯韦-布洛赫方程，拟议的研究包含了广泛的阵列， 理论、计算和应用问题。 这些问题包括 寻找可积Maxwell-Bloch方程的新的显式解， 同宿轨道和混沌动力学，有限维吸引子， 激光器激发态的稳定， 解决方案，以及光纤激光器和二极管激光器的数学描述。 与实际物理和工程应用的比较， 实验也提出了。 教育部分涉及 将作者的研究经验转化为几何和 微分学课程的动态系统导向教学法 大二、大三、大四和研究生水平的方程式，以及 为学生提供建议，并让他们参与与洛杉矶大学的研究合作， 阿拉莫斯国家实验室。 美国国家科学基金会强烈鼓励早期 发展学术教师作为教育工作者和研究人员。 的 教师早期职业发展（CAREER）计划是一个基金会范围内的计划，为初级教师提供支持， 在整体职业发展的背景下。 它结合在一个单一的 计划在最广泛的意义上支持高质量的研究和教育 以及传统上在科学领域代表性不足的人的充分参与 与工程学 该计划增强并强调了 基金会致力于发展全面、均衡的学术 包括研究和教育的职业。 研究部分 这个项目的目的是研究，解决两个定期， 主要是时间周期性的，不规则的，或混乱的行为，在两类 机械和激光光学中的物理系统。 工作重点将放在 近似可积的数学模型，也就是说， 近似的方法离明确地 可解的 通过忽略某些小的量，这些模型确实成为 显式可解或可积。 本文中得到的显式解 方法可以用来近似更复杂的解决方案 近可积系统 因此，拟议的工作将制定一项 数学描述背后的某些类型的机制， 被调查的物理系统的行为，如不规则的 耦合摆振幅的跳动，以及一些规则的和 激光器的混沌运行机制。 将使用数值计算 推动分析调查并确认其结果，以及 为了将他们的结果扩展到不那么简化的数学模型， 因此，不服从显式或近似解，但更 物理上准确。 与实际物理和工程比较 并提出了应用和实验。数学 在调查过程中发现的技术应该是通用的 足以适用于物理学其他领域的类似问题，例如 非线性纤维光学、固体和流体力学。 教育 部分将涉及将作者的研究经验， 大二、大三、大四和研究生阶段的课堂作业， 为学生提供建议，并让他们参与与洛杉矶大学的研究合作， 阿拉莫斯国家实验室。

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.