This is a proposal to study spatially localized structures in driven dissipative systems. Such structures are characterized by a balance between energy input through suitable forcing and energy dissipation within the structure, and are common in many physical systems including fluids, nonlinear optics and reaction-diffusion equations. This proposal seeks to extend existing understanding of these structures to two and three spatial dimensions through mathematical analysis of global bifurcations in spatially reversible systems together with numerical continuation techniques. Systems with both variational and nonvariational structure will be studied, with a focus on understanding the growth and multiplicity of stationary or moving structures as parameters are varied. Time-dependent localized structures will be studied through analysis of the bulk state and the motion of the front or boundary that confines it. Collisions between moving structures and their trapping by external inhomogeneities will be investigated by direct numerical simulation. The results will be applied to several systems exhibiting thermally driven motion including binary fluid convection, doubly diffusive convection and rotating convection.
Spatially localized structures are common in physical systems. Familiar examples in fluids include localized convection, vortices, liquid drops and solitary waves. Additional examples are provided by spots in optical and chemical systems, localized buckling of slender structures under compression, pulses propagating along neural fibers and localized oscillations in vibrating granular media. Although these examples reach across many areas of the physical sciences the localized structures that appear have many properties in common. This proposal seeks to develop a mathematical understanding of the origin and properties of such structures focusing on changes that take place as the parameters of the system change, and aims to identify the properties that are shared by these different examples.
