(a) (b) FIG. 10: The region of secondary stability of traveling waves is marked with asterisks; for details refer to the text. The solid line shows where σ$_{1}$ is purely imaginary; to the right of this line, the traveling waves are certainly unstable. The dashed line shows where q = 91 r$_{2}$/ 144; to the left of this line, traveling waves are also certainly unstable. The dotted lines show where σ$_{2}$ = 0. 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 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. 91 r$_{2}$/ 144 < q < 11 r$_{2}$/ 12 . (30) 20