| Dimensionality reduction techniques have long been used in a number of fields including nonlinear optics and fluid dynamics. Regardless of the specific technique, the underlying idea is to generate a reduced order model that approximates the dynamics of the system but with fewer degrees of freedom. In the field of nonlinear optics, two of the most popular techniques are coupled mode theory and the variational reduction; both of which are based on a solution ansatz. In other fields however, data driven techniques are more common. These techniques extract a set of basis functions from a training set of data and do not require an explicit solution ansatz. In this thesis, we perform a study of the application of dimensionality reduction techniques, both ansatz based and data driven, to problems in nonlinear optics and other physical disciplines. Specifically, we demonstrate that for pattern forming systems, which are ubiquitous in optics, reduced order models can be used to quickly and accurately compute solution branches, even for bifurcation sequences as complicated as the multi-pulsing transition in a mode-locked laser or the route to chaos in an unstable semiconductor waveguide array laser. We also show that a significant degree of computational savings can be obtained while evolving a system in time by exploiting a low dimensional representation of the system when possible. Although we focus on nonlinear optics in this thesis, the techniques outlined here can be applied to any pattern forming system whose dynamics are essentially low-dimensional. To demonstrate this, we also apply these techniques to surface water waves to obtain periodic solution branches in a physical context other than optics. |
