 Benjamin-Feir instability

For the complex Ginzburg–Landau equation $A_t=A+(1+i{c}_1)A_{xx}-(1+i{c}_3)|A{|}^{2}A$, the uniform oscillation has a neutral phase mode whose growth rate is $\lambda =-(1+c_1{c}_3)k^2+O(k^4)$. Negative phase diffusion therefore causes a long-wavelength modulational instability. The Benjamin-Feir stability condition is the complementary positive-diffusion condition. On a finite domain an unstable wavenumber must actually be allowed by the boundary conditions.