Mathematical Sciences: Dynamical Systems Theory Motivated by Bubbles, Accelerators and Split-Operator Numerical Schemes".

小行星9624471 这项工作得到了美国国家科学基金会的支持， 职业发展奖。 研究部分将侧重于 动力系统理论在物理学中的发展和应用 气泡动力学和加速器的问题，以及分裂算子数值格式的研究。 在气泡动力学领域， 将集中在强非线性微分方程建模一- 和双气泡系统。 特别地，提出了双气泡系统的哈密顿公式，揭示了它们的显著特征。 这是一个相互作用的系统，有助于对这些系统进行定量的全球分析。 一个改进的标准孤立的气泡空化也寻求。 在加速器领域，工作包括广泛的分析 以及涉及剪切力非线性增长率的数值问题 层，其饱和振幅的大小， 积分微分方程的Melnikov理论，以及 在一个射频四极杆的调幅光束。 工作 分裂算子方案将涉及识别的显式结构 n阶分块分裂算子格式。 教育计划旨在 将主要研究者的几个方面纳入课堂， 应用动力系统理论。 在研究生一级， 多层次的努力正在进行中，通过目前的研究生在 数学受过训练，进行应用动力系统研究， 来自其他学科的学生（机械工程，认知和 神经科学和经济学）从动力系统中获得工具， 他们的研究。 这项工作还包括开发一门新课程， 气泡振荡的非线性动力学， 奇异摄动理论，和一个完整的修订， 先进 应用数学方法本科课程。 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。

9624471 Kaper This work is supported by a National Science Foundation Faculty Early Career Development Award. The research component will focus on the development and application of dynamical systems theory to physical problems in bubble dynamics and accelerators, as well as the study of split- operator numerical schemes. In the area of bubble dynamics, the research will focus on the strongly nonlinear differential equations modeling one- and two-bubble systems. In particular, a Hamiltonian formulation of two- bubble systems is proposed that reveals the salient features of their interaction and facilitates a quantitative global analysis of these systems. An improved criterion for the cavitation of isolated bubbles is also sought. In the area of accelerators, the work comprises a broad array of analytical and numerical questions that involve the rates of nonlinear growth of shear layers, the magnitudes of their saturation amplitudes, the development of Melnikov theory for integro-differential equations, and the dynamics of amplitude-modulated beams in a Radio Frequency Quadrapole. The work on split-operator schemes will involve identifying the explicit structure of Nth order partitioned split-operator schemes. The education plan aims to incorporate into the classroom several aspects of the principal investigator's experience in applied dynamical systems theory. On the graduate level, a multi-tiered effort is underway through which current graduate students in mathematics are trained to conduct applied dynamical systems research, and students from other disciplines (mechanical engineering, cognitive and neural science, and economics) acquire tools from dynamical systems for their research. The effort also includes the development of a new course in nonlinear dynamics of bubble oscillations, research group activity in singular perturbation theory, and a complete revision of an advanced undergraduate course in the methods of applied mathematics. 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 which include both research and education.