Professor Wayne will study the long-time behavior of partial differential equations like the Navier-Stokes, Euler, and Maxwell's equations which arise in the study of fluid dynamics and optics. He will use methods and techniques from dynamical systems theory to make qualitative and quantitative predictions about the behavior of the solutions of these equations and will focus on five main areas: (i) The derivation, justification and experimental validation of model equations for waves on fluid surfaces; (ii) The long-time behavior of solutions of the Navier-Stokes equations; particularly vortex solutions; (iii) The modeling of very short pulses in optical media; (iv) The existence and stability of pulse solutions in coupled optical systems; and (v) The approximation of the motion of thin elastic media. In particular, he will attempt to extend the insights gained from understanding the invariant geometrical objects in the phase space of ordinary differential equations to the infinite dimensional setting of partial differential equations, focusing particularly on the case of problems on unbounded spatial domains where the presence of continuous spectrum may cause qualitative differences between the behavior of the finite and infinite dimensional systems.
The types of systems that Professor Wayne will study arise in many applications, including nonlinear optical communication, fluid mechanics and the behavior of elastic materials. For instance, attempts to obtain faster and faster transmission of information through optical fibers often utilize extremely short optical pulses. While there is a well understood and much studied method for approximating the transmission of long pulses through such fibers optical technology is now reaching the point where this approximation breaks down and new models are needed. Point (iii) in the list of projects above aims to develop such approximations. This problem, like the others that will be studied in this project, is characterized by the fact that while it is impossible to solve the equations governing the system exactly, applications require at least a qualitative understanding of the behavior of solutions. Professor Wayne's research has three goals: First, the derivation of model equations which can be used to approximate the behavior of the true physical system -- whether it arises in optics, fluids, or elsewhere. Secondly, the computation of accurate estimates to control how much the behavior of the true system can deviate from that of the model system, and finally the development of geometrical insights which can provide a qualitative means of understanding and predicting the behavior of this type of complicated physical system even if the actual solution can not be computed.
