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k = 1 + ϵ $^{2}$q and r = ϵ $^{2}$r$_{2}$ , for ϵ = 0 . 25, q = 0 . 02 with r$_{2}$ = 0 . 04. These values of r$_{2}$ and q lie in the unstable region given in Fig. 10, and the simulations support this prediction of instability. 

## VIII. CONCLUSIONS 

We have examined the stability of spatially periodic solutions to the dispersive Niko- laevskiy equation, which is the original model introduced by Nikolaevskiy [1] for seismic waves. The reincorporation of dispersive effects stands in contrast to most studies subse- quent to Nikolaevskiy’s paper. We have shown how the instability of all spatially periodic solutions at the onset of pattern formation in the more-often treated, nondispersive version is modified by the presence of dispersive terms. Our results have been achieved through both a numerical calculation of the secondary stability boundary for the traveling wave solutions and an asymptotic treatment of three particular scalings in ϵ for the dispersive terms. The secondary stability diagrams (“Busse balloons”) can be rather complicated, and can depend sensitively on the size of the dispersive terms. 

Our consideration of the case α, β = O (1) can be interpreted as giving information about 

