= Three-to-one spatially forced amplitude equation
{title2=$A_T=\mu A-c|A|^2A+F\overline A^{\,2}$}
A critical pattern <Fourier mode> transforms under a spatial translation as $A\mapsto e^{i\phi}A$. A forcing at three times the critical <wavenumber> transforms as $F\mapsto e^{3i\phi}F$. Their lowest resonant product with phase weight one is $F\overline A^{\,2}$. This determines the form of the leading weak-forcing <amplitude equation>, while projection onto an <adjoint eigenfunction> determines its coefficient. Symmetry permits this coupling but does not ensure that it is nonzero. Travelling forcing gives a time-dependent phase to $F$.
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