B. Traveling waves close to the marginal curve VII. NUMERICAL SIMULATIONS OF THE DISPERSIVE NIKOLAEVSKIY EQUATION 21 We now turn to the second case in which a 2 $_{0}$q may be small: in the region close to the marginal stability curve. Following an analysis similar to that for the dissipative Nikolaevskiy equation [4], we find that, in contrast to the dissipative case (in which a narrow region of stable rolls exists close to the marginal curve [4]), here all traveling waves are unstable near the marginal curve. To illustrate some of the consequences of the results of the preceding sections, we have carried out numerical simulations of the dispersive Nikolaevskiy equation, using a pseu- dospectral method, with exponential time stepping [17], of which a small sample are pre- sented here. The initial condition is taken to be a traveling wave with a given wavenumber k (approximated as a cosine of the amplitude given by (8)), plus small random noise, and the domain size is D = 100 π/k . Figure 11 illustrates in order strong (a), intermediate (b) and weak dispersion (c)–(e). The values of α , β and k are chosen in each case to correspond to traveling waves that are predicted to be unstable by the asymptotic analysis. Figure 11 (a) shows the case of strong dispersion, with α = 2 and β = 1, and wave number k = 1 and r = 0 . 01. The stability analysis of (17) predicts that the rolls are unstable, since α = 2 and β = 1 lie in the unstable region in Fig. 5; the numerical simulation agrees with this asymptotic result. In Fig. 11 (b) an example of intermediate dispersion is simulated, where α = 2 ϵ 3 / 4 and β = ϵ 3 / $^{4}$, r = ϵ 2 and k = 1 + ϵq , for q = 0 . 2 and ϵ = 0 . 1. It is known from the asymptotic results of Sec. V that α and β being O ( ϵ 3 / $^{4}$) with wave number k = 1 + ϵq will result in unstable traveling wave solutions, which agrees with the simulation shown in Fig. 11 (b). To show the effects of weak dispersion with wave number k = 1 + ϵ $^{2}$q we take r = 0 . 01 and ϵ = 0 . 1. Figure 11 (c) shows the case α = 2 ϵ , β = 0, q = 0 . 87, while in Fig. 11 (d) the parameter values are α = 0, β = 5 ϵ , q = 2 . 5. Rolls should in each case be unstable, according to the analysis of Sec. VI A, and this is confirmed by the numerical simulations. Figure 11 (e) represents weak dispersion, with α = ϵ and β = 0. The wavenumber is