(a)  0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6 stable stable unstable unstable L q ’ ’ (b)  0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6 stable stable unstable unstable L q ’ ’ FIG. 7: Predicted secondary stability boundaries of spatially periodic solutions of the Nikolaevskiy equation. (a) Nondispersive case ( v ′ = 0), (b) dispersive case ( v ′ = 5). For the case of no dispersion, Tribelsky and Velarde [3] showed that, according to the present scaling, there is monotonic instability of the rolls with q > 0 (with unstable distur- bances having 0 < L < ( a 2 $_{0}$q ) 1 / $^{4}$). By contrast, oscillatory instability occurs for rolls with q < 0 (unstable modes having 0 < L < ( − 2 a 2 $_{0}$q/ 25) 1 / $^{4}$). It is convenient to present our results for the dispersive case in terms of the rescaled variables q ′ = q/r 1 / 2 2 , L ′ = L/r 3 / 8 2 and v ′ = ˆ v/r 3 / 8 2 . Figure 7 illustrates the regions of stability and instability of the traveling waves, in the cases v ′ = 0 and v ′ = 5. Note that in the dispersive case all instabilities are oscillatory. We should view with caution the conclusion above that all traveling waves are unstable, because it relies crucially on the assumption that a 2 $_{0}$q is not small. The stability analysis above breaks down if q or a 2 0 are small; the true stability properties of corresponding traveling waves will be investigated in the next section. In this section, we investigate the cases of traveling waves with wavenumber close to k = 1 or close to the marginal stability boundary, in other words those for which in the previous scaling a 2 $_{0}$q ≪ 1. VI. SECONDARY STABILITY OF TRAVELING WAVES: α, β = O ( ϵ ) A. Traveling waves with close-to-critical wavenumber In order to resolve the secondary stability problem for traveling waves with wavenumber close to k$_{c}$ = 1, we set k = 1 + ϵ $^{2}$q , as was done for the nondispersive case by Tribelsky and 15