= 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 $\phi_T=\lambda\phi_{YY}-\gamma\phi_{YYYY}-\alpha\phi_Y^2\phi_{YY}$, with $\gamma>0$, the straight roll loses transverse <linear stability analysis> when $\lambda<0$. Differentiating gives the conserved slope equation $p_T=\partial_Y^2(\lambda p-\gamma p_{YY}-\alpha p^3/3)$. When $\lambda,\alpha<0$, the nonzero preferred slopes satisfy $p^2=3\lambda/\alpha$.
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