# Properties of Phase Solitons in an Optically Driven Semiconductor Ring Laser
## 3 Results
We see that the equations can be made to depend just on the ratio \(\sigma/b\) by defining the new spatial and temporal variables \(z^{\prime}=bz\) and \(t^{\prime}=bt\)  
\[\frac{\partial E}{\partial z^{\prime}}+\frac{\partial E}{\partial t ^{\prime}} = \frac{\sigma}{b}\left[E_{I}-\left(1+i\theta\right)E+\left(1-i \alpha\right)DE\right]\,, \tag{7}\] \[\frac{\partial D}{\partial t^{\prime}} = \mu-\left(1+\left|E\right|^{2}\right)D\,. \tag{8}\]  
Therefore, the velocity of the phase soliton can depend only on that ratio, while their duration scales with \(b\). More precisely, if we look for a solution of Equations (7) and (8) of the form \(E=E(w)\), \(D=D(w)\) with \(w=zc/v-t\), that is, a pulse propagating at velocity \(v\), we obtain from Equation (7)  
\[E^{\prime}\left(\frac{c}{v}-1\right)=\frac{\sigma}{b}f_{E}\,, \tag{9}\]  
where we have denoted by \(f_{E}\) the content of the square brackets at the right hand side of Equation (7) and by \(E^{\prime}\) the derivative of \(E\) with respect to \(w\). This equation can be made independent of the ratio \(\sigma/b\) by setting  
\[\frac{v}{c}\approx 1-\frac{\Delta\tau}{N\tau_{c}}\,, \tag{5}\]0  
Since \(v\) is very close to \(c\) this is equivalent to \(v/c=1-K_{1}\sigma/b\). From this discussion it follows that the propagation velocity of the phase soliton decreases linearly with the ratio \(\sigma/b=\tau_{e}/\tau_{p}\), while its width is inversely proportional to the parameter \(b=\tau_{R}/\tau_{e}\). Two phenomena concur in slowing down the phase solitons--the inertia of the active medium, which is proportional to \(\tau_{e}\) and the escape of photons from the cavity, which is inversely proportional to \(\tau_{p}\). The width of the phase soliton, instead, is determined only by inertia of the active medium. The above assumptions were fully confirmed by the plots shown in Figure 6a,b, where we collected all the results obtained by varying independently \(\sigma\) and \(b\). The numerical data admit the linear fits  
\[\frac{v}{c}\approx 1-\frac{\Delta\tau}{N\tau_{c}}\,, \tag{5}\]1  
The constants \(K_{1}\) and \(K_{2}\) depend on all the parameters of the system. Here we found \(K_{1}=0.03908\pm 0.00006\) and \(K_{2}=1.48285\pm 0.00266\). The above consideration, however, do not allow to determine the range of values of \(\sigma\) and \(b\) for which the phase solitons exist and are stable. A complete analysis is beyond the scope of this paper but from our simulations we can at least state that in order for the phase solitons to be stable the ratio \(\sigma/b\) must be sufficiently small, that is, we must be in the class-A laser limit. For fixed \(\sigma=0.3\) the smallest value of \(b\) for which we found a stable phase soliton is stable is \(b=5\), that is, \(\sigma/b=0.06\). If the medium is too slow, the laser cannot support stable phase solitons. Assuming that the roundtrip time is \(\tau_{c}=3.6\) ns as in the experiment of Reference [11] this means that the phase soliton is stable for \(\tau_{e}\leq 0.72\) ns. Figure 7a-e shows that before loosing stability the phase soliton changes considerably with respect to the typical shape shown in \[\frac{v}{c}\approx 1-\frac{\Delta\tau}{N\tau_{c}}\,, \tag{5}\]2 Here the soliton is much wider and slower and now its trajectory in the complex plane never touches the stable fixed point \(A\). Panel (f) of the same figure shows the process of destabilization of the phase soliton as we lower the parameter \(b\) to \(b=4\) starting from the stable solution of \(b=5\). The electric field evolves in a complex dynamical state over 2000 roundtrip times. \[\frac{v}{c}\approx 1-\frac{\Delta\tau}{N\tau_{c}}\,, \tag{5}\]3 (**a**) Plot of the phase soliton velocity as a function of \(\sigma/b\). (**b**) Plot of the temporal width of the phase soliton scaled to the roundtrip time as a function of \(1/b\). Both plots admit a linear fit, shown as a red line.