Spatial heterogeneities are often treated as a nuisance in reaction--diffusion theory: they break translation invariance, complicate spectral analysis, and can destroy the clean bifurcation pictures familiar from homogeneous media. In this talk we argue that heterogeneity is not only an obstacle, but also an opportunity providing a controlled way to create, select, and stabilize coherent structures. We present an analytical framework for existence, stability, and bifurcations of solutions in reaction--diffusion systems with spatially varying coefficients, with an emphasis on front and wave-train type dynamics. The discussion is organized around two representative examples. First, we consider heterogeneous front solutions in a FitzHugh--Nagumo equation, where spatial variability induces fronts that move with variable speed through motionless heterogeneous background states. Second, we study wave trains described by a Ginzburg--Landau amplitude equation that arise as slow modulations in a Swift--Hohenberg model, focusing on how nonuniform coefficients shape wave-number selection. A unifying theme is the use of perturbation methods based on a small parameter - but crucially, the parameter does not measure the size of the spatial heterogeneity. Instead, it captures a scale separation or proximity to a critical regime, allowing heterogeneities of order one while retaining analytical tractability. The main novelty is an extension of established perturbative frameworks to systematically incorporate non-autonomous terms, yielding solvability conditions and reduced modulation equations in spatially varying settings.
This is joint work parly with Jolien Kamphuis and partly with Frits Veerman and Lara van Vianen.