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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 

