-0.4 -0.3 -0.2 -0.1  0  0.1  0.2  0  0.5  1 r k FIG. 1: Plot of λ$_{r}$ for the case r = 0 . 1: note the linearly growing modes with wavenumber around k$_{c}$ = 1, and the weakly damped large-scale modes close to k = 0. of the dispersive terms. However, there is a symmetry x ↦→ − x , u ↦→ − u , α ↦→ − α , β ↦→ − β ; as a consequence of this symmetry we need consider only the case β ≥ 0. Linearization around the steady state u ≡ 0 yields the dispersion relation for Fourier modes proportional to e i kx + $^{λt}$. Thus in general these perturbations take the form of traveling waves, with phase speed λ = k 2 [ r − ( k 2 − 1) $^{2}$] + i k $^{3}$( k $^{2}$β − α ) and group velocity The real part of the growth rate, λ$_{r}$ , is plotted in Fig. 1, for r just above the threshold value r$_{c}$ = 0 for the onset of instability. This figure shows that there exists a band around the critical wavenumber k$_{c}$ = 1 of linearly growing modes, and a neutral mode at k = 0 (the so-called “Goldstone mode”), which significantly affects the nonlinear dynamics of (2). Just beyond the onset of instability of the zero solution, it is straightforward to carry out a weakly nonlinear analysis of (2), with This analysis reveals that there are traveling-wave solutions of the form 4 u ∼ ϵa$_{0}$ e i k ( x − st ) + c.c. , (7) r = ϵ $^{2}$r$_{2}$ . (6) c$_{g}$ = − ∂λ$_{i}$ ∂k = k $^{2}$(3 α − 5 βk $^{2}$) . (5) c$_{p}$ = − λ$_{i}$ k = k $^{2}$( α − k $^{2}$β ) (4)