Quasi-Normal Form for a Ring Model of Pump-Coupled Lasers

We study the dynamics of the closed chain of a large number of coupled lasers. The coupling between elements is supposed to be unidirectional. The distributed integro-differential model is proposed which takes into account the delay due to the optoelectronic conversion of signals. The bifurcation value of the coupling coefficient is obtained, at which the stationary state of elements in the chain becomes unstable. It is shown that the critical case has infinite dimension if the number of elements in the chain tends to infinity. A two-dimensional complex Ginzburg-Landau equation with convection is obtained as a quasi-normal form. We get the homogeneous periodic solutions of the quasi-normal which correspond to inhomogeneous traveling waves in a distributed model. Such solutions can be interpreted as phase-synchronized regimes in the chain of coupled lasers.