= Solution
In the <linear complex Ginzburg-Landau equation>, take $U,c_d,\mu$ real. \b[$U$ gives <advection>], carrying the envelope downstream at speed $U$. \b[$c_d$ gives <wave dispersion>]: the imaginary part of the second-derivative coefficient changes wave phase without directly dissipating a real-<wavenumber> <Fourier mode>. \b[$\mu$ gives local linear production or damping]: a spatially uniform perturbation grows for $\mu>0$ and decays for $\mu<0$.
The real part of the diffusion coefficient is the scaled value one. It smooths the envelope and damps a real-<wavenumber> <Fourier mode> at rate $k^2$. Thus the <linear complex Ginzburg-Landau equation> models competing <advection>, smoothing through a <diffusion equation>, <wave dispersion> and local growth, rather than nonlinear saturation.
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