  We are interested in the description of small modulations in time and space
of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*}
\partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi
|\Psi|^2, \end{align*} near the Eckhaus boundary, that is, when the wave train
is near the threshold of its first instability. Depending on the parameters $
\alpha $, $ \beta $ a number of modulation equations can be derived, such as
the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau
based amplitude equations. Here we establish error estimates showing that the
KdV approximation makes correct predictions in a certain parameter regime. Our
proof is based on energy estimates and exploits the conservation law structure
of the critical mode. In order to improve linear damping we work in spaces of
analytic functions.
