| Abstract: |
| Spatial heterogeneities are often treated as an inconvenience: they break translation invariance, complicate spectral analysis, and blur bifurcation scenarios familiar from homogeneous media. In this talk, we argue that heterogeneity is not just a complication, but also a useful tool for generating, selecting, and stabilizing coherent structures. We develop an analytical framework for existence, stability, and bifurcations in reaction-diffusion systems with spatially dependent coefficients, focusing on front and wave train dynamics. The presentation is centered around two case studies. First, we examine heterogeneous front solutions in a FitzHugh-Nagumo equation, where spatial variability produces fronts that propagate at non-constant speeds through stationary heterogeneous background states. Second, we study wave trains governed by a Ginzburg-Landau amplitude equation arising as slow modulations in a Swift-Hohenberg model, showing how spatially non-uniform coefficients affect wave-number selection. A common thread is the use of perturbation techniques involving a small parameter. Importantly, this parameter does not measure the size of the spatial heterogeneity. Instead, it reflects scale separation or nearness to a critical regime, allowing order-one heterogeneities while preserving analytical tractability. The key novelty is extending classical perturbative approaches to include non-autonomous terms systematically. This yields solvability conditions and reduced modulation equations in spatially varying environments naturally. |
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