Long-wave phase reduction of the complex Ginzburg-Landau equation

Write $A=e^{-ict}(1+r)e^{i\varphi }$. The amplitude relaxes at rate two, while the phase is neutral under constant shifts. Eliminating the damped amplitude on long scales gives ${\varphi }_t=(1+bc){\varphi }_{xx}+(c-b)({\varphi }_x{)}^2-E{\varphi }_{xxxx}+\cdots$, 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.