One-dimensional pattern formation with galilean invariance near a stationary bifurcation

One-dimensional pattern formation with Galilean symmetry is not governed by the Ginzburg-Landau equation near onset; an additional equation describing a large-scale mean flow is required. We derive these amplitude equations and predict the solution amplitude. All steady patterns are unstable, with the growth of the instability taking place on a faster time scale than the formation of the pattern itself. Numerical simulations show that chaotic solutions are obtained, whose amplitude is consistent with our theory.