(a) 0.95 1 1.05 0 0.002 0.004 0.006 0.008 0.01 k  r (b) 0.8 1 1.2 0 0.05 0.1 0.15 0.2 0.25 k  r (c) 0.96 0.98 1 1.02 1.04 0 0.002 0.004 0.006 0.008 0.01 k  r FIG. 2: The secondary stability regions of traveling waves of (2), calculated numerically for (a) α = 1 / 2, (b) α = 2 and (c) α = 5, all for β = 0. Shown are the marginal curve r = (1 − k $^{2}$)2 (solid line) and the secondary stability boundary of the traveling waves (dashed line), with stability between the dashed lines. (a) 0.95 1 1.05 0 0.002 0.004 0.006 0.008 0.01 k  r (b) 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 s k  r FIG. 3: The secondary stability regions of traveling waves of (2), calculated numerically for (a) β = 5, (b) β = 5 . 5, both with α = 40. Shown are the marginal curve r = (1 − k $^{2}$)2 (solid line) and the secondary stability boundaries of the traveling waves (dashed line), with stability between the dashed lines. To clarify the regions of stability/instability, the “s” indicates one of the stable regions. of stable waves in the ( k, r ) plane. The stable region is a thin strip, confined to small values of r ; in this case, for r > 0 . 0078 all rolls are unstable. For larger α this strip of stable waves is longer and wider; for example at α = 2 there are some stable rolls up to r ≈ 0 . 22, and at α = 5 the stability region extends at least as far as r = 0 . 9. Furthermore, it is apparent for α = 5 that a symmetrical Eckhaus-like stability region is present for very small values of r (from the numerical results themselves, it seems to be present in all three cases, but is visible only in the last plot of Fig. 2). The shrinkage of the region of stable traveling waves for small α is consistent with there being no stable rolls at all in the nondispersive case. While an exhaustive examination of the secondary stability diagrams across ( α, β ) pa- rameter space is infeasible, it is worthy of note that these diagrams may be extremely 7