(a) 0.95 1 1.05 0 2 4 6 8 10 k (b) 0.8 0.9 1 1.1 1.2 0 2 4 6 8 10 k FIG. 4: The secondary stability of traveling waves of (2) calculated numerically for (a) fixed β = 0 and r = 0 . 01 in ( k, α ) parameter space, (b) fixed α = 40 and r = 0 . 1 in ( k, β ) parameter space. The marginal curve is represented by the solid lines; traveling waves are stable inside the dashed lines. complicated. A good example arises if we set α = 40 and vary β — see Fig. 3. For β = 5 there is a small Eckhaus-like stability region for r < 0 . 001; for larger values of r , there re- mains a single stability region. For larger β , however, the stability region splits into several parts; for example, at β = 5 . 5 there may be up to five separate intervals of stable traveling waves for a given value of r . Above we have presented our secondary stability diagrams in the ( k, r ) plane, for fixed values of α and β . If our interest is in the effects of dispersion on the stability of traveling waves then it is more instructive instead to fix r and present results in either the ( k, α ) or the ( k, β ) plane. Our first example is for r = 0 . 01 and β = 0 — see Fig. 4(a). Given this value of r , the traveling waves exist for 0 . 9487 < k < 1 . 0488. We expect that if α is small enough then all roll solutions are unstable; this is indeed the case. For larger values of α , a region of stable rolls appears. In Fig. 4(b), we show a second case, where we fix α = 40 and r = 0 . 1, to emphasize that the structure of the stability region may be rather complicated, exhibiting a sensitive parameter dependence. In this and the following two sections, we analyse the secondary stability of traveling waves (7). The most straightforward case arises when the dispersion parameters α and β are each O (1). To contrast with later sections, we shall characterize this case as strong dispersion . Whereas the nondispersive Nikolaevskiy equation has no stable spatially periodic states, IV. SECONDARY STABILITY OF TRAVELING WAVES: α, β = O (1) 8