The study of dispersive waves and their dynamics and stability in nonlinear media is fundamental in applications arising from nonlinear optics and mean-field theories in atomic physics. The presence of nonlinearity often requires a combination of asymptotic and perturbation methods, scientific computation, and rigorous mathematical analysis to achieve a solid mathematical framework for understanding a given physical system. By direct collaboration with both industrial partners and members of the physics and electrical engineering academic community, quantitative models for the nonlinear optics and atomic systems of interest will be developed based upon first principles. These models will be studied in appropriate parameter regimes where simplified nonlinear dynamical systems theory can be applied. The results will then be recast in terms of their original experimental context so that the theoretical predictions can be tested, verified, and modified as necessary. The specific applications of interest concern optical parametric oscillators, optical fiber lasers and devices, and Bose-Einstein condensates. All these systems exhibit a stable evolution of nonlinear pulses, fronts, and periodic wavetrains.
This research in mathematical modeling and analysis addresses three classes of important problems. With rapidly developing materials and devices, nonlinear optics remains at the forefront of enabling technologies for communications and information systems. Of primary importance is the stabilization of optical pulses. This research aims to provide a general description of the stability of pulses in a wide variety of lasers and devices where nonlinearity plays a key role. Optical parametric oscillators have tremendous potential application for tunable coherent radiation, pattern recognition, and optical information processing. This work will establish regions of control for the pulse and front structures in these optical devices, facilitating practical implementation of the technology. Bose-Einstein condensates, which have been only recently realized experimentally, are expected to have applications in quantum logic and matter-wave transport. Trapping the condensate and sustaining it for long time periods are fundamental for making the Bose-Einstein condensates a viable technology. This research will focus on various periodic trap configurations that can stabilize the condensate in both attractive and repulsive states.
