Coherent Structures: Interaction and Propagation of Defects

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We will investigate coherent structures in complex systems using a model-independent, dynamical systems approach. Among the phenomena that we will study are the spread of instabilities, the dynamics of wave trains in excitable and oscillatory media, line and corner defects in two-dimensional media, and the interaction of coherent structures. We exploit a detailed description of spectra to separate properties of defects from the background field. We isolate organizing centers where properties of the medium change and bifurcations occur.Important applications range from biology through material science to fluid dynamics and turbulence modeling. Prominent examples are spiral waves in cardiac tissue, spiking and bursting in nerve axons, giant vortices in superconductors, interface propagation in crystallization processes, or crack tip motion. In many of these examples, little is known about the exact microscopic mechanisms, or the mere size of the system does not allow for an accurate prediction of macroscopic behavior from microscopic laws. Coherent structures are the major obstruction to crude statistical approaches in complex systems. We intend to explore the properties of coherent structures using analytical methods and we project to develop reliable numerical tools. The results will further our ability to predict the behavior of complex systems in parameter ranges far from equilibrium by joining large-scale computations, based on a mean field theory, with small-scale information, based on a combined analytical and numerical description of coherent structures.