Nonlinear Dynamics in Models of Wave Propagation and Domain Coarsening

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The aim of this work is to understand significant aspects of wave phenomena through developing and applying the methods of nonlinear dynamics --- the mathematics of systems that change in time and space. A main goal is to explain the robustness or stability of solitons, an important class of waves which exhibit particle-like properties and are capable of a remarkably deep and explicit mathematical description. The mathematics of wave dynamics is also unexpectedly important for prograss concerning a fundamental problem in materials science. To understand how an alloy's microscopic structure coarsens in time, the investigator is developing a new mathematical understanding of wave-like transport for singular mass distributions.This work seeks to explain mathematically a number of poorly understood characteristics of waves of various kinds, surface waves on fluids being a prime example. Such characteristics include: the extraordinary stability of solitons (waves that persist in isolation); the tsunami-like qualities of wakes produced in certain conditions by modern fast car ferries; and the nature of the powerful forces created when waves focus and break simultaneously. The common thread that runs through these investigations is the effective understanding of physical phenomena through the development of new mathematical methods for analyzing dynamic change.