  Order parameter equations, such as the complex Swift-Hohenberg (CSH)
equation, offer a simplified and universal description that hold close to an
instability threshold. The universality of the description refers to the fact
that the same kind of instability produces the same order parameter equation.
In the case of lasers, the instability usually corresponds to the emitting
threshold, and the CSH equation can be obtained from the Maxwell-Bloch (MB)
equations for a class C laser with small detuning. In this paper we numerically
check the validity of the CSH equation as an approximation of the MB equations,
taking into account that its terms are of different asymptotic order, and that,
despite of having been systematically overlooked in the literature, this fact
is essential in order to correctly capture the weakly nonlinear dynamics of the
MB. The approximate distance to threshold range for which the CSH equation
holds is also estimated.
