the bottom of the secondary stability diagram obtained in ( k, r ) parameter space for fixed α and β . Two cases were found: either all traveling waves are unstable at the bottom of the diagram, or there is a symmetrical, Eckhaus-like region of stable traveling waves, right down to onset at r = 0 (although the width of the region of stable rolls does not stand in the usual Eckhaus ratio to the width of the existence region of rolls). The separate analysis for smaller values of α, β can be interpreted as shedding light on the upper parts of the fixed- α , β stability diagram in k, r parameter space. We have shown that for small α, β , a narrow region of stable waves may exist near k = 1. However, beyond the range of validity of the asymptotic analysis, the numerical stability results show the complicated nature of the secondary stability boundaries, so we are unable to draw any significant general conclusions about the form of the secondary stability diagram, limiting ourselves to some specific examples. Things are further complicated by the fact that rolls predicted to be stable by the asymptotics may in fact turn out to be unstable when the full numerical calculation is performed, since the asymptotics concerns only long-wavelength instabilities, and other, short-wavelength instabilities may turn out to be present. In this paper, we have said little about the behavior of time-dependent solutions of the dispersive Nikolaevskiy equation. However, it appears from our numerical simulations that when all waves are unstable, chaotic states are found that have a similar behavior to that found in the non-dispersive Nikolaevskiy equation [5–7]. [1] V. N. Nikolaevskiy, in Recent Advances in Engineering Science , edited by S. L. Koh and C. G. Speziale (Springer–Verlag, Berlin, 1989), no. 39 in Lecture Notes in Engineering, pp. 210–221. [2] H. Fujisaka and T. Yamada, Prog. Theor. Phys. 106 , 315 (2001). [3] M. I. Tribelsky and M. G. Velarde, Phys. Rev. E 54 , 4973 (1996). [4] S. M. Cox and P. C. Matthews, Phys. Rev. E 76 , 056202 (2007). [5] P. C. Matthews and S. M. Cox, Phys. Rev. E 62 , R1473 (2000). [6] M. I. Tribelsky and K. Tsuboi, Phys. Rev. Lett. 76 , 1631 (1996). [7] H. Sakaguchi and D. Tanaka, Phys. Rev. E 76 , 025201 (2007). [8] R. W. Wittenberg and K.-F. Poon, Phys. Rev. E 79 , 56225 (2009). [9] B. A. Malomed, Phys. Rev. A 45 , 1009 (1992). 23