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With real $k,U$ and $\mu >k^2$, substitution into the nonlinear Ginzburg-Landau equation cancels the material derivative and gives $Q^2=\mu -k^2$. A complex $k$ makes the modulus spatially varying, so it generally does not admit this constant-amplitude nonlinear Ginzburg-Landau equation ansatz.

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 isFor $R>0$, $Y=R^2$ obeys the logistic differential equation $Y^{\prime }=2Y(a-Y)$. Equivalently, $Z=1/Y$ solves $Z^{\prime }=-2aZ+2$, whence $Z(t)=a^{-1}+(R_0^{-2}-a^{-1})e^{-2at}$. The nonconstant cubic amplitude saturation solution isIt 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.

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 givesThe stipulated positive-amplitude Ginzburg-Landau plane wave therefore hasThe 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.