content_image/337059.jpg content_image/337064.jpg content_image/337072.jpg content_image/337073.jpg content_image/337071.jpg content_image/337070.jpg
<img src='content_image/337059.jpg'>

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 

and group velocity 

<img src='content_image/337064.jpg'>

<img src='content_image/337072.jpg'>

<img src='content_image/337073.jpg'>

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 

<img src='content_image/337071.jpg'>

This analysis reveals that there are traveling-wave solutions of the form 

<img src='content_image/337070.jpg'>

