Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 331 / 3 / b / i / 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 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.