Self-similar behavior is evident in many physical systems including the rupture of thin films, the pinching of thin liquid threads or solid necks, and the spreading of liquid drops. This research seeks to increase the understanding of self-similarity by viewing the mathematical models for these phenomena in the context of dynamical systems theory. Self-similar solutions can be thought of as steady equilibria in a rescaled coordinate system that is "zooming" to match the scale of the phenomena. Most of these models have an infinite number of self-similar solutions. The foremost goal of this project is the development of a stability theory that will identify which solution is dynamically selected by the model.
The ideas developed in this project will provide mathematicians, engineers and physicists with a set of tools for analyzing self-similar dynamics. A phenomenon is self-similar if it appears to be unchanging once the time scale is suitably accelerated or slowed down and the spatial scale is also adjusted. For example, integrated circuits rely on the deposition of thin metallic traces on a substrate; this research will help determine when these traces will break during fabrication and elucidate the details of how the break occurs, since there is often self-similar behavior near the critical event. Thin fluid films are created in the application of paint, adhesives and optical coatings; the techniques developed in this project should describe when and how films rupture and help devise strategies for preventing these defects, which often develop in a self-similar way. This grant will also underwrite the research of Harvey Mudd College undergraduates who will become an integral part of this investigation.
