where D = 2 π/k is the length of the domain and k is the wavenumber of the solution under consideration. The expression (10) follows from multiplying (2) by f $^{′}$( z ) and integrating over the domain, using integration by parts multiple times. To compensate for the additional unknown c , we have an additional equation from the fact that we may choose the phase of the wave, for example by specifying that ¯ u$_{1}$ is real. After calculating the solution, we construct the eigenvalue problem for perturbations. If we suppose that u ( x, t ) = f ( z ) + ˜ u ( x, t ), then substitution in (2) yields the linearized perturbation equation ∂ ˜ u ∂t = − ∂ 2 ∂x 2 [ r − ( 1 + ∂ 2 ∂x 2 ) 2 ] ˜ u + α ∂ $^{3}$˜ u ∂x 3 + β ∂ $^{5}$˜ u ∂x 5 − f ( z ) ∂ ˜ u ∂x − ˜ uf $^{′}$( z ) . (11) We take ˜ u = e σt +i pz N/ 2 ∑ − N/ 2+1 v$_{n}$ e i $^{nkz}$, where all possible eigenfunctions may be captured by limiting consideration to − k/ 2 ≤ p ≤ k/ 2. The resulting eigenvalue equations to determine the growth rate σ are then ( σ − i cK$_{n}$ ) v$_{n}$ = L v$_{n}$ − N/ 2 ∑ − N/ 2+1 i K$_{m}$v$_{m}$ ¯ u$_{n}$$_{−}$$_{m}$ − N/ 2 ∑ − N/ 2+1 i mkv$_{n}$$_{−}$$_{m}$ ¯ u$_{m}$, where L = K 2 $_{n}$( r − (1 − K 2 $_{n}$) $^{2}$) − i αK 3 n + i βK 5 n and K$_{n}$ = p + nk . The eigenvalues of this system are computed numerically. By examining the largest real part of all eigenvalues σ for a large sample of values of p in the relevant interval, we determine whether the original traveling waves are stable or unstable. In the following section we provide some stability diagrams based on the above method. In determining our results, we have been careful to check that: adequate samples in p are taken (too few, particularly for small values of p , can lead one to miss certain small regions of instability); adequate Fourier modes are taken in determining both the original solution and the perturbations; adequate samples are taken in parameter space to determine all regions of stable rolls. Typically, 300 values of p are used, with N = 16. B. Results Now we present the secondary stability diagrams. The first case considered here is setting β = 0 and varying α — see Fig. 2. When α is small ( α = 1 / 2), there is a very small region 6