This project concentrates on nearly-integrable nonlinear wave equations, that is, integrable partial differential equations and their perturbations. One of the goals is to use a complement of pure mathematical theory, perturbation theory, and numerical experiments to describe, understand, and predict the behavior of these systems. A second goal is to apply these tools and concepts to study specific systems that arise in nonlinear optics. A third goal is to use nearly integrable systems as a model for formulating questions of a statistical nature about the physics of nonlinear interactions and nonlinear mode selection mechanisms. The theme here is to focus on solutions, first the exact integrable solutions and then their behavior under perturbations. The integrable theory yields explicit solutions and the full set of parameters for each N-mode solution, stability within the integrable system, and the nearby solutions that result from unstable solutions. Then numerical experiments on the perturbed systems test whether nearly-integrable dynamics can be approximated by modulated N-mode integrable solutions. If so, the corresponding perturbed equations are derived. These tools are brought to bear on specific applied projects. Integrable systems of partial differential equations often have solutions whose properties can be exactly established. This is quite unlike the usual case for nonlinear partial differential equations. What is important here is that often the properties remain when the system is perturbed. Thus one can tell a great deal about otherwise intractable problems. Such problems arise in nonlinear optics (for example, in the design of long-distance communication networks using fiber optics and the design of optical cavities for very fast switches), in the motion of waves in nonlinear acoustic media, and in flows around thin wings.
