Dynamical Systems Methods for Partial Differential Equations

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Abstract for DMS-1311553Professor Wayne will study how methods such as invariant manifold theory and the Kolmogorov-Arnold-Moser (KAM) theory, which were originally developed to understand finite dimensional dynamical systems, can be adapted to yield insight into the qualitative and quantitative behavior of solutions of partial differential equations. He will concentrate primarily on equations arising in physical applications such as fundamental fluid equations, e.g the Navier-Stokes equation, the equations for vortex sheets, and equations from nonlinear optics. His research will focus on four main problems: (A) Metastable behavior in two-dimensional fluids; (B) Periodic solutions of the vortex sheet equations; (C) Breathers in periodic media; and (D) Normal forms and invariant manifolds in dispersive Hamiltonian systems. Dynamical systems methods have yielded many insights into the qualitative behavior of finite dimensional systems for which no closed form solutions exist. The existence theory of many of the infinite dimensional systems to be studied is now well established and this research will attempt to derive more detailed information about the behavior of the solutions on various physically relevant time scales, as well as illuminating the origin of these time scales.The differential equations that Professor Wayne will study arise in a variety of different physical contexts and are characterized by the fact that while the equations themselves are well known, they are too complicated to solve explicitly except in very special or physically unrealistic cases. Nevertheless, applications require at least a qualitative understanding of the behavior of their solutions and this research project will aim to develop such an understanding for the systems described above. As an example related to point (C) in the preceding paragraph, consider light pulses of the types that are used in fiber optic cables currently used for telecommunications. In a homogeneous medium, such as glass, such pulses rapidly spread out or disperse. However, in periodic media, like certain crystals, such pulses may become trapped. Trapped pulses are of great current interest because of the hope that they might serve as the basis for a purely optical computational system. Professor Wayne's research will examine the conditions that the medium must satisfy in order to support these ``trapped'' pulses as well as investigating how common such media are likely to be. The other three projects will also aim to develop new fundamental insights into these important physical systems.