 Three-to-one spatially forced amplitude equation

A critical pattern Fourier mode transforms under a spatial translation as $A\mapsto e^{i\varphi }A$. A forcing at three times the critical wavenumber transforms as $F\mapsto e^{3i\varphi }F$. 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 $F$.