preclude our making general statements; instead we consider some illustrative special cases. To present the conclusions most generally, it is helpful to introduce q ′ = q/r$_{2}$ , L ′ = L/r 1 / 2 2 , α ′ = ˆ α/r 1 / 2 2 , β ′ = ˆ β/r 1 / 2 2 and v ′ = ˆ v/r 1 / 2 2 . 

Let us begin by considering the special case β ′ = 0. Figure 8 shows where traveling waves with different values of q ′ are stable and unstable to perturbations with wavenumbers L $^{′}$; each panel in the figure corresponds to a different choice of α $^{′}$. In understanding the sequence of transitions in the topology of the various panels, it is helpful to first consider the behavior of the stability boundaries for small α ′ (and hence small v $^{′}$), in particular in the L ′ = 0 limit. We have seen above that the right-hand stability curve (labeled R ) intersects the q ′ axis at q ′ + = 11 / 12 + v ′ $^{2}$/ 1152. For small ˆ v , it follows from (29) that the left-hand stability curve (labeled T ) intersects the q ′ axis at q ′ − ∼ 91 / 144 + 5 | v $^{′}$| / 26568 1 / $^{2}$. Thus as α ′ is increased from zero, q ′ − moves to the right more rapidly than does q ′ $_{+}$. Eventually, at some sufficiently large value of α $^{′}$, q ′ − = q ′ $_{+}$, and all traveling waves are unstable in the limit L ′ = 0. On the other hand, when α ′ is large, q ′ − halts at q ′ − = 131 / 144. However, q ′ + continues to increase, and this results in the appearance of a small- L ′ stability region. In fact, for sufficiently large α $^{′}$, some rolls are stable to disturbances for all L $^{′}$. For 0 ≤ α ′ < α ′ $_{c}$, where α ′ c ≈ 5 . 7, all traveling waves are unstable. For α ′ > α ′ $_{c}$, a stable region appears (see Fig. 8(e)). Subsequently, for any value of α ′ > α ′ c the stable region becomes more apparent. This result can be compared with the numerical stability results shown in Fig. 2(a), where α = 1 / 2. The stability condition α ′ > α ′ c ≈ 5 . 7 (where α ′ = $_{α/}$√$_{r}$$_{) corresponds to}$ r < ( α/ 5 . 7) 2 = 0 . 0077, showing remarkably good agreement with the upper limit of the stable region in Fig. 2(a). 

If instead we consider the special case α ′ = 0, with β ′ > 0, we find a broadly similar picture, in that all traveling waves are unstable when β ′ is small, but some eventually stabilize, once β ′ is sufficiently large. From Fig. 9 it is apparent that the two stability boundaries R and T intersect, coalesce, then lift off from the q ′ axis as β ′ is increased. Ultimately they re-attach to the q ′ axis, when β ′ = β ′ $_{c}$, where β ′ c ≈ 5 . 06 as shown in Fig. 9(g). For β ′ > β ′ $_{c}$, there is a region of stable traveling waves. 

Let us now express the results above in a form more illuminating for comparison with our earlier numerical secondary stability calculations (Sec. III). As an example, we set ˆ α = 1 and ˆ β = 0 and consider the limit of small L , looking for regions of stable waves as r$_{2}$ is varied. 

