Critical Phenomena in Coherent Structure Formation

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Large complex dynamical systems often self-organize into simple coherent behavior. Describing, analyzing, and computing the dynamics of these self-organized structures is at the center of understanding processes ranging from crystallization, self-assembly, or material hardening, to the emergence of patterns and function in living organisms. The principal investigator (PI) studies the role of coherent structures both in organizing the dynamics of systems through the selection of patterns and shape, but also in their role for validating models through quantitative predictions. The first part of the project is concerned with invasion processes, where a new structure propagates in the system after a sudden change in parameters. This invasion process is poorly understood and computationally challenging. The PI will develop new analytical and computational tools to systematically analyze, predict, and thus validate models. The second part of the project is concerned with point defects that arise in soft matter. Point defects both organize the system dynamics, but also present challenges to macroscopic, homogenized descriptions of large systems. The proposed work further develops analytical and computational methods that aim at a fine description of the shape of the core of point defects and their effect on the medium far away from the core. Graduate and undergraduate students will be engaged in the research of the project.The PI will analyze invasion fronts and point defects. Invasion fronts describe the propagation into an unstable state across many scientific areas, from material science to ecology and epidemiology. Propagation speeds and the novel state created in the wake of the invasion process can often be well predicted from linear analysis at the unstable state, through a marginal pointwise stability analysis. The PI will first develop novel computational tools for the associated linear questions, complemented by a rigorous convergence analysis. In a second step, the PI will addresse the validity of these linear predictions in nonlinear systems, in particular questions associated with the marginal stability conjecture that postulates that critical, only marginally stable fronts are selected in the invasion process. A second part of the proposal studies dislocations in striped phases and creases in elastic media. In both situations, analysis of the existence of these point defects will yield a deeper understanding of the role of the core of the defect in the large-scale deformation of the medium. The interaction between core and farfield is analyzed using analytical, perturbative tools and computational methods that rely on rigorous approximations of unbounded by finite-size domains.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.