= Solution
Put $a=\mu-k^2=Q^2>0$. The shared travelling phase again cancels <advection> in the <nonlinear Ginzburg-Landau equation>; the remaining amplitude <ordinary differential equation> is
$$
R'=aR-R^3.
$$
For $R>0$, $Y=R^2$ obeys the <logistic differential equation> $Y'=2Y(a-Y)$. Equivalently, $Z=1/Y$ solves $Z'=-2aZ+2$, whence $Z(t)=a^{-1}+(R_0^{-2}-a^{-1})e^{-2at}$. The nonconstant <cubic amplitude saturation> solution is
$$
\boxed{R(t)=\frac{Q}{\sqrt{1+\left(Q^2/R_0^2-1\right)e^{-2Q^2t}}},\qquad t\geq0.}
$$
It takes the specified $R_0$ at $t=0$. The denominator of the <cubic amplitude saturation> solution remains positive for all future time: if $R_0<Q$ its bracket exceeds one, and if $R_0>Q$ it lies between $Q^2/R_0^2$ and one. Thus the positive solution is global forward in time. $R_0=Q$ would give the excluded constant solution; $R_0=0$ would remain at zero rather than relax to the positive wave.
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