| In the first part of this thesis, we consider a system of two Schrödinger equations coupled together through the nonlinearity; models of this sort have been developed to describe the effects of birefringence in an optical fiber as well as the incoherent interaction of two optical beams in a slab. Mathematically, localized pulses in a fiber and beams in a slab are considered as standing waves. Such waves are important as information carriers—and the profiles of these waves and their stability upon perturbation are of great interest. We describe a family of multi-component pulses, as well as multi-component N-pulses, that bifurcate from a simple one-component stationary wave as a system parameter is increased. We also develop numerical and analytical tools to analyze the stability of these waves. It is shown that the bifurcating multi-component pulses are stable for a range of parameters near the bifurcation point and a geometric mechanism is provided that can spur an eventual instability. A related criterion is used to show that all of the bifurcating multi-component N-pulses are unstable.; In the second part of this thesis, we consider a model for pulse propagation in the regime of strong dispersion management; this model takes the form of a Schrödinger equation with a nonlocal nonlinearity. We approximate and study dispersion managed solitons using their characterization as minima of an averaged variational principle. This approach helps to explain the persistence of the dispersion managed soliton in the regime of negative residual dispersion and the mechanism for its disappearance as the residual dispersion decreases further. We also describe the discovery of a bisoliton that has implications for increased data transmission rates and more advanced coding schemes. |
