SPATIOTEMPORAL INSTABILITIES OF LASERS IN MODELS REDUCED VIA CENTER MANIFOLD TECHNIQUES

Several models of partial differential equations describing the dynamics of lasers with transverse effects are introduced by using the center manifold theorem for the elimination of irrelevant variables. By taking advantages of the different time scales associated with the relaxations of the variables, we first eliminate the polarization and later the population inversion. We show that in contrast with the plane-wave models, the use of center manifold techniques is necessary to properly describe the spatiotemporal behaviors of lasers. In particular, we characterize unsuspected Hopf bifurcations for the model obtained from the adiabatic elimination of the polarization and discuss the presence of diffusive terms induced by the interaction of radiation and matter after the elimination of the population inversion. A complex Ginzburg-Landau equation is also obtained in the small-field limit.