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

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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.