= Benjamin-Feir stability condition
{c}
For $A_t=A+(1+ib)A_{xx}-(1+ic)|A|^2A$, the uniform oscillation $A=e^{-ict}$ has phase <diffusion> coefficient $D=1+bc$. Its amplitude-phase <dispersion relation> is $\lambda^2+(2+2k^2)\lambda+2Dk^2+(1+b^2)k^4=0$. 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 $k^2<-2D/(1+b^2)$.
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