not hold, and requires reconsideration. We should expect such analysis to break down in this limit, because Fig. 5 is inconsistent with the known behavior of the nondispersive Nikolaevskiy equation ( α = β = 0), for which all rolls are unstable at onset [3, 5]. Thus in the next section we consider smaller values of α, β . It turns out, after some experimentation, that small α and β first lead to a new scaling if we adapt the scaling first used by Tribelsky and Velarde [3] for the nondispersive case, and extended by Cox and Matthews [4] to a damped version of the Nikolaevskiy equation. In this scaling the original traveling waves remain O ( ϵ ), but the perturbation to the traveling- wave amplitude is O ( ϵ 3 / $^{2}$) and the large-scale mode is O ( ϵ 7 / $^{4}$); furthermore, slow spatial and temporal variations of perturbations take place on scales given by X = ϵ 3 / $^{4}$x , T = ϵ 3 / $^{2}$t , τ = ϵ 3 / $^{4}$t . (Note that these slow variables are different from those of the previous section, but our notation for slow variables is consistent within sections.) To allow the development of consistent amplitude equations for the perturbation we then take ˆ c = ˆ α − ˆ β and ˆ v = 3ˆ α − 5 ˆ β . Here a ( X, T ) represents disturbances to the amplitude of the pattern, ψ ( X, T ) represents corresponding disturbances to the phase of the pattern and f ( X, T ) is a large-scale mode. Substitution of u , as given by (20), in (2) requires the consideration of the problem at successive orders in ϵ 1 / $^{4}$. After much consequent algebra, we find the (nonlinear) amplitude equations 13 ∂ψ ∂T = 4 ∂ $^{2}$ψ ∂X 2 − f − ˆ v ∂ψ ∂X , ∂f ∂T = ∂ $^{2}$f ∂X 2 − 2 a$_{0}$ ∂a ∂X , ∂a ∂T = 4 ∂ $^{2}$a ∂X 2 − 4 a$_{0}$ ( ∂ψ ∂X ) 2 − 8 a$_{0}$q ∂ψ ∂X − ˆ v ∂a ∂X . M = (1 + ϵq ) x − ˆ cτ − ϵ 1 / $^{4}$ˆ vqT + ϵ 1 / $^{4}$ψ ( X, T ) , u = ϵ ( a$_{0}$ + ϵ 1 / $^{2}$a ( X, T ))e i M + c.c. + ϵ 7 / $^{4}$f ( X, T ) + · · · , (20) α = ϵ 3 / $^{4}$ˆ α, β = ϵ 3 / 4 ˆ β. Applying a weakly nonlinear analysis to (2) gives where a$_{0}$ = 6 √ r$_{2}$ − 4 q $^{2}$, V. SECONDARY STABILITY OF TRAVELING WAVES: α, β = O ( ϵ 3 / $^{4}$)