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

数学科学:近可积方程的微扰理论及其应用

基本信息

  • 批准号:
    9502142
  • 负责人:
  • 金额:
    $ 23.41万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    1995
  • 资助国家:
    美国
  • 起止时间:
    1995-07-15 至 2001-06-30
  • 项目状态:
    已结题

项目摘要

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.
9502142 Kovacic这项工作得到了国家科学基金会早期职业发展奖的支持。 该研究将集中于近综合系统的理论。这些系统是完全集成的普通和部分微分方程或截然不同的方程式的小扰动。将解决两个领域:低维系统中的多脉冲同质轨道,以及描述环形激光光学器件的Maxwell-Bloch integro-Partial差微分方程的常规和不规则动力学。在多脉冲同型轨道领域提出的研究是作者以前在不稳定共振系统上的工作的延续。它将在大型近乎综合系统的大家庭中展示几类新类的多脉冲轨道,因此揭示了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,用于验证这些复杂的同质轨道的存在,因此在特定示例中,因此不规则的动力学。 还提出了这些方法在力学,流体和固体动力学以及非线性光学方面的应用。在Maxwell-Bloch方程的区域中,提出的研究包含一系列的理论,计算和应用问题。 这些问题包括找到可集成的Maxwell-Bloch方程的新的明确解决方案,同型轨道和混乱动力学,有限维的吸引子,激光态的稳定态,解决方案的数值模拟以及纤维激光器和DIODE激光器的数学描述。 还提出了与现实的物理和工程应用和实验的比较。 该教育部分涉及将作者的研究经验转化为几何和动态系统的方法,以二年级,初中和研究生级别的微分方程教学课程进行教学课程,并向学生提供建议,并向他们提供与Los Alamos国家实验室的研究合作。 国家科学基金会强烈鼓励学术教师作为教育者和研究人员的早期发展。 教师早期职业发展(职业)计划是一个基础范围的计划,在整体职业发展的背景下为初级教师提供了支持。 它在一个计划中结合了最广泛的质量研究和教育的支持,以及传统上代表性不足的科学和工程专业人士的全部参与。 该计划增强并强调了基础对包括研究和教育在内的全面,平衡学术职业的发展的重要性。 该项目的研究组成部分涉及预期的研究,该研究涉及在机械和激光光学方面两类物理系统的规则,主要周期和不规则或混乱的行为。 这项工作将集中于几乎可以综合的数学模型,即,近似程度的模型距离使它们明确解决是一小步的。 通过忽略某些少量,这些模型确实可以明确解决或可以集成。 以这种方式获得的明确解决方案可用于近似更复杂的近乎整合系统的解决方案。 因此,拟议的工作将对正在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,例如耦合摆的振幅中的不规则节拍,以及某些激光的常规和混乱的操作制度。 数值计算将用于激励分析研究并确认其发现,并将其结果扩展到较少简化且因此不适合显式或近似解决方案但在物理上更准确的数学模型。 还提出了与现实的物理和工程应用和实验的比较。在本研究过程中发现的数学技术应足够一般,以适用于其他物理学领域的类似问题,例如非线性光纤,固体和流体机械。 该教育部分将涉及将作者的研究经验纳入大二,初中和研究生级的课堂工作,并向学生提供建议,并参与与Los Alamos国家实验室的研究合作。

项目成果

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Gregor Kovacic其他文献

Gregor Kovacic的其他文献

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{{ truncateString('Gregor Kovacic', 18)}}的其他基金

OP: Collaborative Research: Nonlinear Theory of Slow Light
OP:合作研究:慢光非线性理论
  • 批准号:
    1615859
  • 财政年份:
    2016
  • 资助金额:
    $ 23.41万
  • 项目类别:
    Standard Grant
Dynamics of Light Interacting with Active Media
光与活性介质相互作用的动力学
  • 批准号:
    1009453
  • 财政年份:
    2010
  • 资助金额:
    $ 23.41万
  • 项目类别:
    Standard Grant
MSM: Collaborative Research: Cortical Processing Across Multiple Scales
MSM:协作研究:跨多个尺度的皮层处理
  • 批准号:
    0506287
  • 财政年份:
    2005
  • 资助金额:
    $ 23.41万
  • 项目类别:
    Standard Grant
Mathematical Modeling of the Visual Cortex
视觉皮层的数学建模
  • 批准号:
    0308943
  • 财政年份:
    2003
  • 资助金额:
    $ 23.41万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Applied Dynamics of Near Integrable Systems
数学科学:近可积系统的应用动力学
  • 批准号:
    9403750
  • 财政年份:
    1994
  • 资助金额:
    $ 23.41万
  • 项目类别:
    Standard Grant

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4D Robust Optimization in Intensity-Modulated Proton Therapy
调强质子治疗中的 4D 鲁棒优化
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