The simplest solutions of CGLE are plane wave solutions, which take the form of a one-parameter family:
where
We examine the stability of these solutions by setting
which, after substitution into the CGLE equation and linearising, yields
We now set
after which, a little manipulation gives
We seek solutions to these equations in the form 
which results in a polynomial for 
. We
then expand the solutions for for
small 
(the long-wavelength limit) to find
Thus, travelling waves solutions are long-wave stable as long as the condition
holds. Hence, when the Benjamin-Feir-Newell criterion
holds, there is a spectrum of wavenumbers
including the band centre (
) state,
where plane waves are stable.
, the stability condition reduces
to the Eckhaus condition 
.
, the Benjamin-Feir
instability has a non-zero group velocity 
and so the instability is of a convective nature.