Spatial Localization in Several Dimensions

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This research project will study spatially localized patterns such as fronts, spots, or pulses that arise in fluids, nonlinear optics, chemical catalysis, and other continuum systems. Examples include vortices, drops, and solitary waves in fluids, spots in optical and chemical systems, localized buckling of slender structures under compression, pulses propagating along nerve fibers, and localized oscillations in vibrating granular media. These diverse systems have two things in common: (i) they dissipate energy and need to be sustained by external forcing, and (ii) they are sensitive to this external forcing and small changes may lead to distinctly different outcomes (patterns). This project seeks to extend existing theory to similar systems in two and three dimensions and to provide a comprehensive understanding of the mechanisms responsible for growth of patterns when the forcing is varied. It will apply the results to improved understanding of vortices in rotating convection and crystal growth from a melt. The project will involve a graduate student in the research. Spatially localized structures are common in solutions to many partial differential equations arising in physical modeling. This research project studies dissipative systems driven by periodic forcing in which different forcing amplitudes can lead to distinct localized states. The project seeks to extend existing theory to higher dimensions and to provide a comprehensive understanding of the mechanisms behind the different types of growth of the structures that are observed when parameters are varied. The structures of interest include not only spatially localized pulse-like states, but also fronts connecting two distinct states and defects or holes in otherwise spatially periodic structures. In addition, the project aims to apply the results to several systems of importance in physics and engineering, including vortex structures in rotating convection and crystallization from a supercooled liquid. The techniques used include bifurcation theory for reversible and near-reversible systems, coupled with numerical branch-following and direct numerical simulations of realistic systems.