Scale disparities in the complex Swift-Hohenberg equation for lasers.
The complex Swift-Hohenberg (CSH) equation is a generic order parameter equation that applies to many physical systems. In the case of class C lasers, it can be obtained from the Maxwell-Bloch equations using the assumptions of slow envelope and small detuning. We show that the resulting CSH equation inevitably contains different asymptotic order terms, associated with the dominance of the effect of dispersion over diffusion. These scale disparities are usually overlooked or simply not mentioned in the literature, assuming that a CSH equation with all terms of the same order still provides qualitative information. In this paper, the asymptotically nonuniform CSH equation is carefully deduced using a simpler scaling-free procedure, and a stability analysis of the simplest solutions together with some numerical simulations are presented, in which the mentioned scale disparities are clearly seen.