= Phase modulation
{title2=$w(x,y,t)=w_0(x+\phi(y,t))+\cdots$}
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 $\xi=x+\phi(Y)$ must be differentiated with $\mathcal D_Y=\partial_Y+\phi_Y\partial_\xi$. Omitting this <chain rule> can change the nonlinear coefficient of a reduced <amplitude equation> even when its linear coefficients are unchanged.
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