Stability and Dynamics of Dispersive Waves in Nonlinear Media

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Nonlinear dispersive waves arise in a diverse set of application fields. The dynamics and stability of these waves is of paramount importance to understanding the underlying physical properties and behavior of a given physical system. Using an interdisciplinary approach that combines asymptotic and perturbation methods, scientific computation, and rigorous mathematical analysis with models which are based on experimental observations of nonlinear phenomena, a fundamental understanding can be achieved of specific optical and atomic systems. In conjunction with the modeling efforts, the mathematical objectives are to further develop and extend modern methods utilized for quantifying and understanding the wave dynamics of nonlinear, dispersive partial differential equations. In particular, a variety of methods for reducing the governing equations to more easily handled partial and ordinary differential equation systems is pursued. Of specific interest to all the atomic and optical systems considered here is the stability and persistence of localized solutions that often result from soliton-type solutions of some underlying Hamiltonian (integrable) system.The stability of localized solutions, or pulses, is of fundamental importance in optical and atomic physics. In optical physics, the stability of pulse solutions is critical for determining the operating regimes of the so-called, mode-locked laser. In the past decade, mode-locking technologies have gone from a fundamental science to a commercially viable technology with applications in imaging, medical sciences, and telecommunications. Characterizing, improving, and understanding the operational limits of these lasers is a central focus of the proposal. Additionally, emerging photonic technologies are predicated on the existence and stability of pulse structures as they form the basis for optical bits in all-optical signal processing and switching. A more fundamental investigation of pulse stability lies in the area of atomic physics where Bose-Einstein condensates have realized experimentally the existence of matter waves. Such matter waves have a direct analog to mode-locking technologies, allowing for the possibility of creating a pulsed matter laser.