 Zigzag instability

A zigzag instability is a long-wave transverse phase modulation of parallel rolls. In ${\varphi }_T=\lambda {\varphi }_{YY}-\gamma {\varphi }_{YYYY}-\alpha {\varphi }_Y^2{\varphi }_{YY}$, 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 $p^2=3\lambda /\alpha$.