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Assume $\alpha >0$, so a finite nonzero critical wavenumber exists. The linear growth rate of a Fourier mode is $\lambda (k)=-\alpha +\mu k^2-k^4$. Maximizing over $k$ and setting that maximum to zero givesSet $K=k_c^2=\sqrt{\alpha }$. The resonant three-wave triad has wavevectors satisfy ${\mathbf{k}}_1+{\mathbf{k}}_2+{\mathbf{k}}_3=0$ and ${\mathbf{k}}_i\cdot {\mathbf{k}}_j=-K/2$ for $i\ne j$. With the slow time and amplitude scaling, every growth, quadratic and cubic term first enters at order ${ϵ}^3$. Projection onto each critical mode removes the nonresonant correction ${\Theta }_3$ by the method of multiple scales solvability condition.

The parameter displacement contributes ${\mu }_{2}KA$ to the first mode. For the quadratic term, use $\nabla \cdot ({\Theta }_1\nabla {\Theta }_1)=\frac{1}{2}{\nabla }^2({\Theta }_1^2)$. The resonant coefficient of ${\Theta }_1^2$ at ${\mathbf{k}}_1$ is $2\bar{B}\bar{C}$, so its contribution is $+\gamma K\bar{B}\bar{C}$.

For the cubic gradient term, an ordered triple of modes with wavevectors $\mathbf{p},\mathbf{q},\mathbf{r}$ summing to ${\mathbf{k}}_1$ contributes $(\mathbf{p}\cdot \mathbf{q})({\mathbf{k}}_1\cdot \mathbf{r})$ times its amplitude product. The three self-interaction permutations yield $-3K^2|A{|}^{2}A$. The six permutations of $({\mathbf{k}}_1,{\mathbf{k}}_2,-{\mathbf{k}}_2)$ yield $-2K^2-4({\mathbf{k}}_1\cdot {\mathbf{k}}_2{)}^2=-3K^2$, and likewise for the third mode. Therefore the hexagonal convection amplitude equations areFor a nonzero convection roll, $B=C=0$ and $|A{|}^2={\mu }_2/(3K)$, requiring ${\mu }_2>0$. Choose its phase so $A=a>0$. The radial perturbation decays, but a perturbation in the two other orientations givesThe eigenvalues are $\pm |\gamma |Ka$. ThusThis is roll instability from broken up-down symmetry. A growing homogeneous amplitude perturbation already proves instability, without a sideband calculation. The source's slightly asymmetric setting means nonzero quadratic coupling. If $\gamma =0$ is allowed literally, the displayed leading equations instead have a continuous sphere of constant-total-intensity states and neutral orientation perturbations. They then establish marginality, not strict growing-mode instability; higher-order effects would decide further selection. Likewise $\alpha =0$ would invalidate the assumed finite-$k_c$ triad scaling.