Spatiotemporal chaos with strong scale separation

Numerous physical, chemical and biological systems display the phenomenon of pattern formation: as a control parameter passes a critical value, a spatially uniform equilibrium becomes unstable to a stable nonuniform, finite-wavelength pattern. The scaling and stability of these patterned solutions beyond the instability threshold is by now well-understood via the Ginzburg-Landau partial differential equation, which describes modulations of the pattern (in one space dimension).

I will discuss the profound effects of an additional continuous symmetry on pattern formation by describing recent results on the Nikolaevskiy equation, a paradigmatic (6th-order) PDE model for short-wave pattern formation with Galilean invariance. In this case, all periodic steady states are unstable beyond the instability threshold, and one observes a direct transition from spatial homogeneity to spatiotemporal chaos with strong scale separation. The asymptotically consistent modulation equation description associated to this PDE, proposed by Matthews and Cox, predicts scaling different from the usual Ginzburg-Landau scaling. I will present some of the novel dynamical and statistical behaviors of these PDE models. These include anomalous scaling of the large-scale mode inconsistent with the Matthews-Cox prediction (but captured by higher-order corrections to the amplitude equations), localized coexistence of ordered and chaotic states, coarsening to domain size-dependent fronts, and stabilization of Burgers-like fronts by spatiotemporal chaos.