= Solution
For this nonlinear ansatz, take $k$ real as well as $U,\mu$ real. Then $|e^{ik(x-Ut)}|=1$, and the advective time <derivative> vanishes. Substitution into the <nonlinear Ginzburg-Landau equation> gives
$$
0=(\mu-k^2)Q-Q^3.
$$
The stipulated positive-amplitude <Ginzburg-Landau plane wave> therefore has
$$
\boxed{Q=\sqrt{\mu-k^2}.}
$$
The strict inequality $\mu>k^2$ guarantees that this is real and positive. The zero solution of the <nonlinear Ginzburg-Landau equation> also exists but is not the required positive wave. Although complex $k$ is useful in the linear impulse analysis, it cannot generally be carried into this constant-amplitude nonlinear ansatz: its spatially varying modulus would make the cubic term carry a different spatial factor.
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