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 finite amplitude 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.
In this project the PI studied spatially localized states in two and three spatial dimensions. Such states are stationary solutions of the nonlinear equations that describe spatially extended systems arising frequently in fluid mechanics, physical chemistry and biology and take the form of spots, rings, vortices and localized oscillations called oscillons. Earlier work by the PI described the universal structure, now referred to as the snakes-and-ladders structure, of the parameter region containing one-dimensional localized states of different lengths as well as different bound states of such states. This parameter region thus contains a very large multiplicity of different localized states, many of which can be simultaneously stable. The PI has identified similar states in a variety of fluid systems, including binary convection (figs 1 & 2), convection in a fluid layer rotating uniformly about the vertical and convection in an imposed vertical magnetic field, and showed that these are organized is a similar snakes-and-ladders structure. The latter two exhibit a snakes-and-ladders structure that is slanted, implying that localized states are present over a broad parameter range. This behavior is a consequence of a conserved quantity whose presence modifies the standard snakes-and-ladders structure. In addition, the PI has identified new types of three-dimensional localized states in binary convection in porous media that resemble snowflakes. In the standard snakes-and-ladders structure localized one-dimensional states grow in length by adding cells at either end. The PI has identified a new type of growth mechanism, called defect-mediated snaking, in which the localized state grows by the splitting of the cell in the center. Model equations, such as the Swift-Hohenberg equation (figs 3-5), can be used to make qualitative predictions of the behavior found in the much more complex systems that arise in applications, a property that was exploited by the PI to generate moving localized states and to study their subsequent collisions.
