= Solution
When $R\ll Q$, the ratio of cubic damping to linear amplification is $R^2/Q^2\ll1$. The <cubic amplitude saturation> equation therefore gives
$$
\frac{R'}R=Q^2-R^2\simeq Q^2=\mu-k^2,
\qquad
R(t)\simeq R(t_0)e^{(\mu-k^2)(t-t_0)}
$$
while the small-amplitude regime lasts. With $c_d=0$, the matching real-<wavenumber> <normal mode> of the <linear complex Ginzburg-Landau equation> has $\omega=Uk+i(\mu-k^2)$ and hence amplitude proportional to $e^{(\mu-k^2)t}$. Thus
$$
\boxed{\gamma_{\mathrm{small\ amplitude}}=\gamma_{\mathrm{linear\ Fourier\ mode}}=\mu-k^2.}
$$
This is the <temporal growth rate> of the same Fourier component. It is not the fixed-position <absolute frequency> rate $\mu-U^2/4$ of a localized impulse. Nonlinear damping becomes important when $R$ is comparable with $Q$ and then arrests exponential amplification.
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