The principal physical impetus for the projects studied in this proposal is to gain an analytical understanding of pattern formation. In particular the PI seeks to characterize the classes of defects that appear when one is far above the parameter threshold at which patterns emerge. These issues are studied through a non-convex variational problem known as the regularized Cross-Newell (RCN) model. Unlike similar models that have been studied recently, RCN incorporates features of non-trivial twist which enable the existence of a richer ?taxonomy? of defects (in particular, concave and convex disclinations). Such features are indeed seen in experiments and numerical simulations of the underlying microscopic physical equations. The issue has been to determine whether or not the variational model can capture these defects. In a recent work the PI has demonstrated that the minimizers of a variational model with twist must necessarily differ from those without twist. The main projects of this proposal are centered on establishing a precise analytical characterization of the competition between microstructure formation and overall topologically induced energetic stress in versions of the RCN model having some geometric symmetry.
The study of pattern forming systems is a fundamental area of scientific investigation in which the tools of modern mathematical analysis can be brought to bear on the modeling of physical systems especially near a critical transition in the behavior of the system. For the projects studied in this proposal one is principally interested in patterns that arise when a continuous translational symmetry is reduced, at a critical threshold, to a discrete periodic symmetry resulting in what is often referred to as a "striped" pattern. Such patterns are for instance generic in Rayleigh-Benard convection (RBC) which is a principal theoretical model for the formation of weather patterns. In RBC the striped pattern corresponds to the formation, at a critical temperature, of periodic "convection rolls" of a uniform characteristic width. A major goal of this research is to study not just the patterns that form at a critical threshold but to characterize the types of defects that arise in these patterns when one is far from threshold. This work will make definite predictions on defect formation that can be tested in the laboratory. This will have relevance for modeling defect structure in liquid crystals, in animal coat patterns (including fingerprints) and in the evolution of plant patterns.
This project concerns mathematical models of extended physical systems that spontaneously form patterns when some parameter, on which the system depends, is stressed beyond a critical threshold. A classical example of this is the Rayleigh-Bénard convection (RBC) experiment in which a fluid trapped between two plates is heated from below. Below a critical value of the temperature the fluid does not move; but just above this critical value the fluid does move and within the bulk of the experimental cell one witnesses an alternating periodic pattern of warmer upswelling fluid followed by cooler downflowing fluid. From a distance this convective configuration may be described as a pattern of periodic stripes. The phenomenon has some relation to the physical process that drives cycles of evaporation and condensation in weather patterns; but in the simpler RBC experiment one can study this in a more controlled and systematic way. This pattern forming structure is also seen in many other physical systems ranging from liquid crystals to optical patterns in laser systems. The main focus of this project has been to mathematically model what happens when these systems are stressed further beyond threshold. Physically, structural bifurcations emerge that are characterized by defect formation. This is analogous to what one sees in material (as opposed to fluid) systems when grain boundaries and dislocations form. However, in the fluid systems, far from threshold, the signature point defects are disclinations rather than dislocations. In the basic model we study we have been able to establish rigorous mathematical results that narrow the range of possible defect structures to a small class that includes what is seen in experiments. Moreover, in a simplified geometry (for the experimental cell), we have derived a family of self-consistent solution candidates for the variational equations of our model that match the defect-mediated patterns observed in experiments. Future investigations will be directed toward benchmarking these solutions against controlled physical experiments and extending the analysis to more complicated geometries.
