Phase modulation ($w(x,y,t)=w_0(x+\varphi (y,t))+\cdots$)

A phase modulation changes the local position of a periodic pattern. Its local wavenumber is shifted by derivatives of the phase. In a method of multiple scales, a correction profile depending on the translated coordinate $\xi =x+\varphi (Y)$ must be differentiated with ${\mathcal{D}}_Y={\partial }_Y+{\varphi }_Y{\partial }_{\xi }$. Omitting this chain rule can change the nonlinear coefficient of a reduced amplitude equation even when its linear coefficients are unchanged.

Let $p={\varphi }_Y$, $r={\varphi }_{YY}$ and $w_2=C_1(\xi )r+C_2(\xi )p^2$. Two applications of the chain rule give ${\mathcal{D}}_Y^2{w}_2=C_1{\varphi }_{YYYY}+2(C_1^{\prime }+C_2)p{\varphi }_{YYY}+(C_1^{\prime }+2C_2)r^2+(C_1^{\prime \prime }+5C_2^{\prime })p^{2}r+C_2^{\prime \prime }{p}^4$. In a Fredholm solvability condition, the fourth term generally contributes to the coefficient of ${\varphi }_Y^2{\varphi }_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.

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.