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| Consider a system of one-dimensional reaction-diffusion equations that models a general oscillatory medium. Such a model exhibits coherent structures or defect solutions that are time-periodic in an appropriate co-moving frame and are connecting two spatially periodic traveling waves at each infinity. Source-type solutions are those having their asymptotic group velocities pointing outward away from the core of the defect. As perturbations will be transported by the group velocities, perturbed source solutions will create non-localized responses in the phase dynamics, and hence analyzing nonlinear dynamics near a source solution is technically complicated. I will present a joint work with Beck, Sandstede, and Zumbrun that establishes nonlinear stability of general spectrally stable sources. This involves the introduction of new analytical techniques that both illuminate and unify previous results on modulational stability of shocks and patterns and expand the range of possible future applications. |
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