ar X iv : p hy si cs / 0 51 10 25 v 1 3 N ov 2 00 5 Coexistence of single – mode and multi – longitudinal mode emission in the ring laser model

A homogeneously broadened unidirectonal ring laser can emit in several longitudinal modes for large enough pump and cavity length because of Rabi splitting induced gain. This is the so called Risken-Nummedal–Graham-Haken (RNGH) instability. We investigate numerically the properties of the multi–mode solution. We show that this solution can coexist with the single–mode one, and its stability domain can extend to pump values smaller than the critical pump of the RNGH instability. Morevoer, we show that the multi–mode solution for large pump values is affected by two different instabilities: a pitchfork bifurcation, which preserves phase–locking, and a Hopf bifurcation, which destroys it.