Coarsening to Chaos-Stabilized Fronts

We investigate a model for pattern formation in the presence of Galilean symmetry proposed by Matthews and Cox [Phys.\ Rev.\ E \textbf{62}, R1473 (2000)], which has the form of coupled generalized Burgers and Ginzburg-Landau-type equations. With only the system size $L$ as a parameter, we find distinct "small-$L$" and "large-$L$" regimes exhibiting clear differences in their dynamics and scaling behavior. The long-time statistically stationary state contains a single $L$-dependent front, stabilized globally by spatiotemporally chaotic dynamics localized away from the front. For sufficiently large domains, the transient dynamics include a state consisting of several viscous shock-like structures which coarsens gradually, before collapsing to a single front when one front absorbs the others.