the introduction of a dispersive term can stabilize periodic traveling waves [11]. Our aim in this paper is to provide a systematic examination of the effects of dispersion. By varying the parameters corresponding to dispersion, we can find when dispersion stabi- lizes traveling waves and investigate how the chaotic state in the non-dispersive equation arises as the dispersion is reduced. In the following section we give the form of the equation and the traveling waves under consideration. Computational results on the stability of these waves are given in Sec. III. The stability analysis of the waves is complicated and depends on the magnitude of the dispersion terms; three different scalings are considered in Secs. IV, V and VI. Sec. VII illustrates some numerical simulations of the Nikolaevskiy equation with dispersion, and our conclusions are summarized in Sec. VIII. We examine the Nikolaevskiy equation with dispersion in the form II. THE NIKOLAEVSKIY EQUATION WITH DISPERSION ∂u ∂t = − ∂ 2 ∂x 2 [ r − ( 1 + ∂ 2 ∂x 2 ) 2 ] u − u ∂u ∂x + α ∂ $^{3}$u ∂x 3 + β ∂ $^{5}$u ∂x $_{5}$, (2) u ( x + D, t ) = u ( x, t ) (3) d d t ∫ D 0 u ( x, t ) d x = 0 , where α and β are the dispersion coefficients. This equation is thus the one originally proposed by Nikolaevskiy [1] (and later examined in [9, 10]), with all spatial derivatives up to the sixth appearing on the right-hand side. In the numerical simulations presented in Sec. VII, we shall impose the periodic boundary condition for some domain length D . Before proceeding, we note that (2) has the same Galilean symmetry ( x ↦→ x + V t , u ↦→ u + V ) as the nondispersive equation (1). Moreover, in view of the Galilean symmetry and the observation that, when the boundary condition (3) is imposed, the spatial average of u may be set as zero (by transforming to a moving frame of reference if necessary). The reflection symmetry ( x ↦→ − x , u ↦→ − u ) of (1) is broken by the presence 3