Benjamin-Feir stability condition

For $A_t=A+(1+ib)A_{xx}-(1+ic)|A{|}^{2}A$, the uniform oscillation $A=e^{-ict}$ has phase diffusion coefficient $D=1+bc$. Its amplitude-phase dispersion relation is ${\lambda }^2+(2+2k^2)\lambda +2D{k}^2+(1+b^2)k^4=0$. Thus $D<0$ gives a long-wavelength modulational instability, whereas $D>0$ gives linear stability of all nonzero Fourier modes. On a fixed periodic domain, instability requires an allowed wavenumber with $k^2<-2D/(1+b^2)$.

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.