= Zigzag instability
{title2=$\sigma(k)=-\lambda k^2-\gamma k^4$}
A <zigzag instability> is a long-wave transverse <phase modulation> of parallel rolls. In the model
$$
\phi_T=\lambda\phi_{YY}-\gamma\phi_{YYYY}-\alpha\phi_Y^2\phi_{YY},\qquad \gamma>0,
$$
a small <Fourier mode> has growth rate $\sigma(k)=-\lambda k^2-\gamma k^4$. Thus the straight pattern is linearly stable for $\lambda>0$ and unstable to sufficiently long transverse waves for $\lambda<0$.
When $\lambda<0$, the unstable band is $0<k^2<-\lambda/\gamma$. Differentiating $\sigma$ with respect to $k^2$ gives the fastest growing scale $k^2=-\lambda/(2\gamma)$ and maximum growth $\lambda^2/(4\gamma)$. The fourth derivative stabilizes short waves rather than allowing arbitrarily fine structure to grow fastest.
Writing the slope as $p=\phi_Y$ gives
$$
p_T=\partial_Y^2\left(\lambda p-\gamma p_{YY}-\frac\alpha3p^3\right),
$$
a conserved <Cahn--Hilliard equation> for the local orientation. When $\lambda<0$ and $\alpha<0$, its homogeneous bulk potential $W(p)=\lambda p^2/2-\alpha p^4/12$ has two minima at $p=\pm\sqrt{3\lambda/\alpha}$. These are preferred local slopes; boundary conditions and the conserved mean slope still constrain the complete pattern.
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