(a) (b) (c) FIG. 6: Numerical simulations of the amplitude equation (17): in each case the real part of A is plotted as a function of ξ and T . In (a) and (b) α = 10, whereas in (c) α = 8 . 4; in each case β = 2 . 6. The initial condition in each case is a plane wave, with n wavelengths in the computational box − 32 π < ξ < 32 π , plus small-amplitude random noise: (a) n = 28 (hence q = 0 . 875); (b) n = 10 ( q = 0 . 3125); (c) n = 20 ( q = 0 . 625). 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 12