This study concerns the behavior of solitary-wave (pulse) solutions to certain nonintegrable Nonlinear Schrodinger-type partial differential equations that have drawn attention from physicists and engineers in recent years, particularly in nonlinear optics applications. The study will focus mainly on systems that model pulse propagation in parametric waveguides made of materials with quadratic nonlinearity. These equations share the property that their solutions show highly phase- sensitive behavior (for example in collision outcomes). One part of the study will investigate the existence and stability of travelling waves for certain systems, using variational techniques and detailed eigenvalue analysis of linearized operators. A second part of the study will involve the development of ordinary differential equations as models for phase- sensitive pulse behavior, including collisions between two quadratic pulses.
The equations to be studied arise in fiber optics; their solutions model light pulses travelling down a fiber. They are of interest to optical scientists and engineers because they model light pulses that are unusually narrow and require unusually low energies to produce; the hope is that many such pulses could be packed closely together to increase the speed of data transmission through optical fibers. This study will address fundamental properties of these pulses by identifying factors important for predicting their stability (when a pulse is affected by noise and interactions with the fiber and other pulses, to what extent does it retain the pulse shape necessary to transmit information?) and their behavior in collisions (when two pulses meet each other head-on).
