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regions denoted by “u” show where 1 + a ( b + d ) < 0, and hence all plane waves are unstable. As discussed in Sec. II, only the region β ≥ 0 need be presented. The existence of a stable region in Fig. 5 is consistent with the numerical results of Sec. III; for example Fig. 2 shows a stable region when α = O (1) and β = 0. 

In Fig. 6, we illustrate the considerations above with some numerical simulations of the modified complex Ginzburg–Landau equation (17). Our numerical code is pseudospectral, and uses exponential time differencing [17]. In each case the initial condition is a plane wave plus small-amplitude random noise. For the simulations illustrated in Fig. 6(a) and (b), 1 + a ( b + d ) > 0. The two plots show the fate of initial conditions in the unstable and stable regions of Fig. 5, respectively. In each case, a stable plane wave is obtained at large T . Figure 6(c) shows the development of instability in the case 1 + a ( b + d ) < 0, where all plane waves are unstable. Here the solution is persistently time-dependent. 

The analysis above tells us about the secondary stability of traveling-wave solutions of the dispersive Nikolaevskiy equation when α, β = O (1), and the results are summarized in Fig. 5. We may think of this analysis as holding for any fixed α and β (not both zero) in the limit as r → 0; thus we expect the lowest part of the secondary stability diagram in ( r, k ) parameter space to reflect Fig. 5. 

However, as indicated earlier, when α and β are both small, the analysis above does 

