Zigzag instability ($\sigma (k)=-\lambda k^2-\gamma k^4$)

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 $p^2=3\lambda /\alpha$.

At a transverse phase-diffusion threshold the coefficient of the second spatial derivative is of order ${\epsilon }^2$. Its product with two long-wave derivatives therefore has order ${\epsilon }^4$, the same order as the stabilizing fourth derivative. This balance determines the slow time $T={\epsilon }^{4}t$ and retains both terms in the leading phase modulation equation.