This project aims at a description of dynamics near Turing patterns in spatially extended systems that do not necessarily possess a free energy. Turing-like patterns are spatially periodic and stationary structures that emerge in a self-organized fashion. Typical contexts are reaction-diffusion settings, but also various convection experiments, or nonlinear optical feedback systems. We propose a systematic investigation of perturbations of Turing patterns, bifurcations from Turing patterns, and defects embedded in Turing patterns. We will aim at model-independent descriptions whenever possible, emphasizing topological and geometric intersection theory rather than explicit solutions. Some specific problems are the existence of dislocations in Turing patterns, symmetry-breaking bifurcations from Turing patterns, inhomogeneities and parameter ramps, and connections with Liesegang structures.
Self-organized, regular patterns are striking when they appear in nature, in sand dunes, and on animal coats. They also carry great technological potential when exploited in manufacturing processes at nanoscales. In both cases, one would like to understand when and how regular structures form, and at which characteristic scale. We plan to study those questions using geometric and topological tools in the analysis. Those tools can shed light on the role of inhomogeneities and defects in the selection of patterns in situations where precise models and parameter values are unavailable. Ultimately, we would like to understand those self-organized pattern formation processes from macroscopic, universal principles that are largely independent of the underlying detailed microscopic processes.
Striped patterns appear in nature in a self-organized fashion. Striking examples include animal coat patterns and sand ripples. The simplest controlled chemical experiment that yields striped phases is a precipitation process designed by Liesegang in 1896, involving only two electrolytes in a gel. As one of the electrolytes diffuses into the gel, it forms a precipitate together with the previously dissolved second electrolyte. As the precipitation process progresses through the gel, it develops oscillations that leave behind precipitation bands with characteristic spacing laws. Researchers have been trying to develop theory that allows us to predict and control the formation of such patterns. The results of the proposal explain for the first time both the emergence and robustness of such patterns and give quantitative predictions for observed length scales. They also identify fundamental mechanisms that lead to the emergence of regular patterns when one might well expect noisy, disorganized structures: the creation of patterns in the wake of propagating fronts confines the self-organized dynamics to the narrow front interface and drastically reduces the role of noise in the process. The results have potential impact on our understanding of patterns in biological growth processes. They also guide our understanding of surface instabilities in manufacturing processes. The second part of the proposal initiated a systematic study of possible defects in such striped patterns, starting with mathematical existence proofs and leading to robust numerical methods that may allow us to understand and control defects in a systematic fashion. The mathematical tools developed here allow for a particle-field type decomposition of internal defect properties and its effect on the far-field deformation of the pattern.
