= Solution
Represent the leading real pattern by $A(T)e^{ik_cx}+\overline A(T)e^{-ik_cx}$. Before forcing, continuous <translation symmetry> acts as $A\mapsto e^{i\theta}A$, and spatial <reflection> acts as $A\mapsto\overline A$. Equivariance excludes quadratic terms and permits a linear term and $|A|^2A$ at cubic order. Symmetry under <reflection> makes their coefficients real. A <supercritical bifurcation> has a saturating cubic coefficient; rescaling amplitude and time makes it $-1$. The imperfection has a Fourier component at the same <wavenumber>, so its resonant projection contributes an additive real coefficient. On the spatially uniform <wave envelope> subspace the <amplitude equation> is therefore $A_T=\mu A-|A|^2A+\alpha$. Allowing modulations would additionally produce the spatial terms of the <real Ginzburg–Landau equation>.
For a forcing pattern moving in the positive $x$ direction, its laboratory-frame coefficient is $\alpha e^{-i\Omega T}$, where $\Omega$ is proportional to $k_cc$ after converting to slow time. The comoving amplitude is $B=e^{i\Omega T}A$, since the fast carrier becomes $e^{ik_c(x-ct)}$. Differentiation gives
$$
\boxed{B_T=(\mu+i\Omega)B-|B|^2B+\alpha.}
$$
A constant phase choice makes $\alpha>0$, and <complex conjugation> reverses the sign of $\Omega$, so take $\Lambda=\Omega>0$. Put $B=\alpha^{1/3}a$ and $\tau=\alpha^{2/3}T$. The rescaled linear coefficients are $\mu/\alpha^{2/3}$ and $\Lambda/\alpha^{2/3}$, and the forcing becomes one. Below the rescaled variables are renamed $A,T,\mu,\Lambda$, giving the <directly forced cubic amplitude equation>.
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