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 th
e 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.
The Nikolaevskiy model for pattern formation with continuous symmetry exhibits spatiotemporal chaos with strong scale separation. Extensive numerical investigations of the chaotic attractor reveal unexpected scaling behavior of the long-wave modes. S
urprisingly, the computed amplitude and correlation time scalings are found to differ from the values obtained by asymptotically consistent multiple-scale analysis. However, when higher-order corrections are added to the leading-order theory of Matthews and Cox, the anomalous scaling is recovered.
