Stability and existence of traveling wave solutions of the two-dimensional nonlinear Schrodinger equation and its higher-order generalizations

The two-dimensional cubic nonlinear Schrödinger equation (NLSE) is a partial differential equation (PDE) that can be used as a model of phenomena in physical systems ranging from waves on deep water to pulses in optical fibers. Traveling waves often play an important role in studies of these systems. In this thesis, we study analytically and numerically the stability and existence of traveling wave solutions of the NLSE and its higher-order generalizations.; There are three main sections of this thesis. In the first section, we review the properties of the entire class of one-dimensional, bounded, traveling wave solutions of the NLSE and we prove the existence and construct the form of the one-dimensional, bounded, traveling wave solutions of two higher-order generalizations of the NLSE.; In the second section, we show asymptotically that all one-dimensional, bounded, traveling wave solutions of the two-dimensional NLSE are linearly unstable with respect to long-wave transverse perturbations. We compare these asymptotic results with results from numerical simulations of the two-dimensional NLSE with perturbed one-dimensional traveling wave solutions used as initial conditions. We complete the section by making a prediction for physical experiments of waves on deep water.; In the final section, we introduce two new symplectic PDE integration schemes. The first of these schemes can be used to solve PDEs with nonlinear parts solvable by the method of characteristics. The second scheme can be used to solve PDEs that require three-way splitting. We use these schemes to study the one-dimensional bounded solutions of the PDEs introduced in the first section.