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Note that dispersion is represented in these equations only through the terms ˆ vψ$_{X}$ and ˆ va$_{X}$ , representing advection of the pattern envelope with the group velocity ˆ v . Note also that the group velocity of the large scale mode f is zero, and hence no corresponding term appears in the second of these equations. 

The three amplitude equations may be reduced to the single (nonlinear) phase equation 

<img src='content_image/330390.jpg'>

Then linearising this equation and setting ψ = e i LX + σT yields the dispersion relation 

<img src='content_image/330388.jpg'>

Before considering this dispersion relation for general L , it is helpful to consider the two limiting cases, of small and large L . First, if L is small, then σ 3 ∼ 16 a 2 $_{0}$qL$^{2}$. Thus, to leading order in L , σ = σ$_{2}$$_{/}$$_{3}$L 2 / $^{3}$, where σ 3 2 / 3 = 16 a 2 $_{0}$q ; hence all traveling waves are unstable if L is small. On the other hand, if L is large, then we have σ 3 + 9 σ $^{2}$L2 + 24 σL 4 + 16 L 6 ≈ 0, and so σ ≈ − L 2 or − 4 L 2 (twice); hence traveling waves are stable to large- L disturbances. In summary, all traveling waves are unstable at onset (provided a 2 $_{0}$q ̸ = 0; in fact we shall see later that when a 2 $_{0}$q is suitably small, we shall need to reconsider this conclusion). The rest of the section provides more details of the instability, for general values of L . 

In order to find the secondary stability boundary for the traveling waves, we set σ = iΩ in the dispersion relation (22), where Ω is real. From the real and the imaginary parts, we obtain 

<img src='content_image/330391.jpg'>

and then after eliminating Ω between these two equations we find that this stability boundary is given by 

<img src='content_image/330387.jpg'>

We note that in this equation L and ˆ v appear only as even powers and thus we can restrict our attention to positive L and ˆ v with no loss of generality. However, both even and odd powers of q occur, so no such economy is possible in considering q (indeed, in the light of [3], we should expect different behaviors for q > 0 and q < 0). 

