Global regularity and inertial manifolds for the Moore-Greitzer model of turbo-machine engine and modeling of pulse propagation in optical fibers

This dissertation consists of two different parts. The first part is an analytical study of the Moore-Greitzer model of turbo-machine engine. We study the regularity and long-time behavior of the solutions to the Moore-Greitzer model of turbo-machine engine. In particular, we prove that this dissipative system of evolution equations possesses a global invariant inertial manifold, and therefore its underlying long-time dynamics reduces to that of an ordinary differential system. Furthermore, we show that the solutions of this model belong to a Gevrey class of regularity (real analytic in the spatial variables). As a result, one can show the exponentially fast convergence of the Galerkin approximation method to the exact solution, an evidence of the reliability of the Galerkin method as a computational scheme in this case. The rigorous results presented here justify the readily available low dimensional numerical experiments and control designs for stabilizing certain states and traveling wave solutions for this model.; The second part consists of study of wave propagation in random medium and modeling femtosecond pulse propagation in optical fibers. We conduct a computational study of wave motion in a weakly disordered optical fibers. Specifically, we implement the transparent boundary condition method in the numerical simulation and study interaction between solitons and radiation in the presence of randomly varying fiber parameters. The results of our computational experiments are confirmed by the theoretical predictions. Next, we provide a model which describes nonlinear ultrashort pulse propagation in optical fibers. In particular, the phase signature of the propagation of a soliton, and soliton self-frequency shift are described and validated through experimental observations. Finally, we apply the Genetic Algorithm based pulse shaping technique to femtosecond pulse propagation in single-mode optical fibers. We demonstrate, through simulation, the system evolves towards an optimal filter configuration that, when applied to shape the input pulse, allows successful transmission of the pulse without loss of intensity, in contrast to unfiltered pulse propagation.