  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.
