Nonlinear Ginzburg-Landau equation (${\psi }_t+U{\psi }_x=\mu \psi +{\psi }_{xx}-|\psi {|}^2\psi$)

This nondispersive cubic complex Ginzburg–Landau equation balances linear growth against nonlinear damping. Its real-wavenumber Ginzburg-Landau plane wave has amplitude $\sqrt{\mu -k^2}$ when $\mu >k^2$. Relaxation within that single-mode amplitude family does not by itself establish stability against general perturbations.

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.

For $a>0$, the positive amplitude ordinary differential equation $R^{\prime }=aR-R^3$ has solution $R(t)=\sqrt{a}[1+(a/R_0^2-1)e^{-2at}{]}^{-1/2}$. This follows from the logistic differential equation for $R^2$. Every $R_0>0$ approaches $\sqrt{a}$ as a monotone function, and the positive distance to this limit decreases; an oscillating complex wave carrying this amplitude has no ordinary monotone ordering.