= Long-wave phase reduction of the complex Ginzburg-Landau equation
Write $A=e^{-ict}(1+r)e^{i\phi}$. The amplitude relaxes at rate two, while the phase is neutral under constant shifts. Eliminating the damped amplitude on long scales gives $\phi_t=(1+bc)\phi_{xx}+(c-b)(\phi_x)^2-E\phi_{xxxx}+\cdots$, where the linear phase branch has $E=b^2(1+c^2)/2$. Near $1+bc=0$, $E=(1+b^2)/2$ at leading order. Constant phase <symmetry> forbids undifferentiated phase terms; spatial parity of the leading amplitude equation forbids odd linear derivatives. Balancing negative phase diffusion against the fourth derivative produces the <Kuramoto-Sivashinsky equation>.
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