content_image/342427.jpg content_image/342434.jpg
<img src='content_image/342427.jpg'>

sides of the secondary stability region are no longer approximately parallel. 

Note the qualitative similarity between the shapes of the stable regions in Fig. 10 and Fig. 2. 

The question remains of whether or not this stable region extends to indefinitely large values of r$_{2}$ . To investigate the large- r$_{2}$ behavior of the stability region, we consider large r$_{2}$ with q = O ( r$_{2}$ ), motivated by the observation, from Fig. 10(b), that stable rolls lie in some region between straight lines in ( q, r$_{2}$ ) parameter space. In this limit the stability condition from (28) simplifies to q < 11 r$_{2}$/ 12, while σ$_{2}$ = 91 / 2 − 72 q/r$_{2}$ + O ( r − 1 / 2 2 ). Hence we can conclude that the region of stable waves for small L and large r$_{2}$ is 

<img src='content_image/342434.jpg'>

In summary, the results of this section show that when α and β are O ( ϵ ), there can be a narrow region of stable traveling waves near k = 1, and that there is no upper limit on the size of r$_{2}$ allowing stable rolls. 

For even smaller values of α and β , of order ϵ $^{2}$, we have checked that α and β do not appear in the leading order amplitude equations, so in that case all traveling waves are unstable, as in the non-dispersive case. 

