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

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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.