I. INTRODUCTION In 1989, Nikolaevskiy [1] derived a model for longitudinal seismic waves, in the form of a one-dimensional partial differential equation for a displacement velocity. Although Nikolaevskiy’s equation included dispersive terms, most subsequent analysis has treated a simplified version of the PDE, in which these terms are omitted. This reduced form is now generally known as the Nikolaevskiy equation , which may be written in the form where r is a control parameter. The equation (1) has been proposed as a model for several other physical systems, including phase instabilities in reaction–diffusion equations [2], elec- troconvection [3] and transverse instabilities of fronts [4]. More generally, (1) can be regarded as a simple model of a pattern-forming system with an instability at finite wavenumber and a neutral “Goldstone” mode arising from symmetry [3, 5]. The uniform state u ≡ 0 of (1) becomes unstable at r = 0 to spatially periodic “roll” solutions, with wavenumbers around k = 1. However, these, in turn, are themselves all unstable at onset in sufficiently large domains [3]; this unusual instability arises from the neutral mode at wavenumber k = 0. In fact, numerical simulations show that the Niko- laevskiy equation exhibits spatiotemporal chaos at onset [5, 6]. The scalings associated with this chaotic regime are unusual in pattern forming systems [5, 7], and this interesting feature of the equation has stimulated significant investigation [8]. Although in some applications (such as the instability of fronts [4]) the omission of dis- persive terms is justified on symmetry grounds, this is not the case in the original context of a model for seismic waves [1]. Earlier work that has considered the effects of dispersion includes the paper of Mal- omed [9], who reinstated one dispersive term in the Nikolaevskiy equation and analysed the secondary stability of traveling-wave solutions by means of coupled Ginzburg–Landau- type equations for the amplitude of the traveling waves and a large-scale mode. His results showed that dispersion could stabilize waves; however, his derivation was not entirely asymp- totically self-consistent [3]. Kudryashov and Migita [10] showed, on the basis of numerical simulations, that traveling waves can be stabilized by the presence of dispersive terms in the Nikolaevskiy equation. It is also known that in the related Kuramoto–Sivashinsky equation, 2 ∂u ∂t = − ∂ 2 ∂x 2 [ r − ( 1 + ∂ 2 ∂x 2 ) 2 ] u − u ∂u ∂x , (1)