Numerical validation of the complex Swift-Hohenberg equation for lasers

Abstract

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.