Stability and dynamics of solitary waves in nonlinear optical materials

Two distinct optical systems which are governed by nonlinear wave equations are investigated, and are found to support localized structures, or solitary wave solutions. The first of these physical phenomena is modeled by a system of two equations which are coupled by a rapidly oscillating quadratic nonlinearity. The interaction between diffraction and the quadratic coupling can be balanced such that the effects of diffraction are diminished, and localized solutions are achieved for long propagation distances. In the second physical system, N fields are coupled through a cubic nonlinearity and a saturable gain term. Through a balance between dispersion and cubic nonlinearity, pulse trains can be achieved which are stable to perturbations in various system parameters and initial conditions.; The quadratic system of equations is investigated from two perspectives. The first considers the system from a variational approach. Assuming a gaussian or hyperbolic secant profile, a reduced system of ordinary differential equations is derived which approximates the behavior of the full system. Conditions on initial profiles are determined, for which solutions remain periodic, resonant with oscillation in the nonlinearity, and hence stay localized in the transverse dimension. Stability analysis shows that these solutions are stable subject to perturbations in initial conditions. The reduced system provides a qualitative means of describing the underlying dynamics which govern the physical system, and yet is much simpler than previous descriptions. The same physical system is then studied by means of a multiple scales approach. It is seen that the diffractionless limit, the effect of nonlinearity can be modeled by a reduced system that has analytic solutions which remain localized in the transverse direction, and periodic in the direction of propagation. Solvability conditions for higher order corrections are derived, which determine the transverse structure of desired solutions, as well as their slow-scale behavior.; In the second system to be considered, a new model to describe N-channel mode-locking, is presented. It combines master mode-locking for a single channel, with terms that allows fields to interact via cubic nonlinear coupling, as well as through a saturable gain. Three different gain models are presented, which account for both self-saturation and cross-saturation of the channels. Analytic solutions for single-channel and dual-channel operation are presented, and the stability of these solutions is addressed. It is seen that the stability of solutions depends on the saturable gain model chosen, and that inclusion of self-saturation in the gain is required for stability. The conditions under which these results may be extended to the N-channel case is investigated both analytically and numerically.