# Properties of Phase Solitons in an Optically Driven Semiconductor Ring Laser
## 2 Methods
In the extended system where the dynamical variables are allowed to depend on the longitudinal coordinate \(z\) the homogeneous state \(A\) is still stable but it coexists with the phase soliton, that is, a state in which the phase varies by \(2\pi\). The stability of the phase soliton is checked as usual by adding dynamical noise to the equations. In Reference [13] the stability and the properties of the phase solitons were studied assuming that all the parameters but the amplitude \(E_{I}\) of the injected field are fixed. In other words the different phase solitons that can be observed varying one experimental parameter in a given laser configuration were considered. Here we take a different approach and, by fixing the amplitude \(E_{I}\) of the injected field, we study how the stability and the properties of the phase solitons change for different laser configurations, defined by the fundamental time scales associated with the active material \((\tau_{e})\) and with the cavity \((\tau_{R},\tau_{p})\). Precisely, in the following we set \(E_{I}=0.0445\) and let \(\sigma\) and \(b\) vary, which means that for a fixed cavity length we assume to vary the photon lifetime \(\tau_{R}\) and the carrier lifetime \(\tau_{e}\). Figure 2 shows a typical example of phase soliton obtained for our initial choice of the parameters \(\sigma=0.3\) and \(b=50\)  
Figure 1: Intensity of the stationary states for the choice of the parameters \(\mu=1.1\), \(\alpha=3\), \(\theta=-2.7\), \(\sigma=0.3\) and \(b=50\) as a function of the injected field amplitude \(E_{I}\). The blue curve is the homogeneous solution. The phase solitons, whose maximum and minimum intensity are shown in red, exist in the interval delimited by the dashed vertical lines. In that region the homogeneous solution is triple valued. The upper state \(A\) is stable, the state \(B\) in the negative slope branch is unstable as usual, the lower state \(C\) is an unstable focus. Panel (a) is a space-time intensity plot where the horizontal axis is time scaled to the roundtrip time and the vertical axis counts the number of roundtrips. Here we assume to observe the field intensity at a fixed position inside the cavity (for instance at the exit of the active medium). We could as well imagine to fix time and follow the evolution of the spatial profile of the intensity one roundtrip after the other. In that case, however, the horizontal axis should be reversed, so that an image symmetric with respect to the vertical axis would be obtained. In our false-color plots, the abscissa represents time in a frame moving at velocity \(c\), thus a rightward (leftward) tilted evolution corresponds to \(v<c\) (\(v>c\)). All stable phase solitons propagate at a velocity smaller than \(c\), so that a stable phase soliton is described by a line inclined to the right. The larger is the inclination, the slower is the phase soliton. Panels (b) and (c) show the time evolution of the field intensity and phase, respectively, over the last roundtrip. The phase soliton displays a deep minimum followed by a small maximum in the intensity and a positive \(2\pi\) phase kink in the phase. Panel (d) shows the evolution of the complex electric field during one roundtrip time. The labels \(A\), \(B\) and \(C\) denote the three fixed points of the homogeneous stationary curve. The trajectory starts and terminates in \(A\), which is the stable fixed point on the upper branch of the bistable curve, it is repelled by \(B\), which is the unstable fixed point on the negative slope branch and circles around the origin \(O\) and around \(C\), which is the unstable focus on the lower branch of the bistable curve. The tip of the complex electric field moves anticlockwise along the trajectory, apart for the small portion between \(A\) and \(B\). This motion results in the positive phase kink of \(2\pi\) shown in (c). The phase soliton with a negative phase kink is always unstable [11; 12].