Spatially Localized States in Extended Systems

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Spatially localized structures are common in continuum systems such as fluids, nonlinear optics and the chemical reactions arising in catalysis. Examples are provided by localized convection, spots in optical and chemical systems, localized buckling of slender structures under compression, pulses propagating along neural fibers, oscillons in vibrating granular media, liquid drops, and solitary waves on flowing liquid films. These diverse systems have two things in common: (i) they are dissipative systems driven by homogeneous forcing, and (ii) there is range of forcing within which the application of different finiteamplitude perturbations can lead to distinct localized states. This proposal seeks to extend existing theory, partly developed by the PI and coworkers, to higher dimensions and to provide a comprehensive understanding of the effects of finite domain size, anisotropy and loss of translation invariance on the origin and properties of these structures. The techniques used include bifurcation theory for reversible and near-reversible systems, coupled with numerical branch-following and direct numerical simulations of realistic systems.Many systems respond to spatially uniform forcing by producing a spatial pattern. These patterns may take the form of a periodic array of cells or spots, or in particular cases by producing a single spot or group of spots. This proposal seeks to understand the relation between these two types of response, and to predict the conditions under which the latter response may be expected. There are many potential applications of a spot-like response. In optics individual spots may be "written" and "erased" using a laser beam, a procedure that may be used to store information. Mechanical structures often buckle in a localized way. Thin liquid films may break up into drops. A chemical process that uses catalysis may be degraded because the chemical conversion fails to proceed uniformly in space. These are all examples of spatially localized structures of the type that will be studied as part of this project. These spot-like structures may also move and interact. Such moving structures are involved, for example, in signaling along nerve fibers. The project seeks to predict the formation of these structures and to understand their basic properties.