# Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quinic Nonlinearity
## 4 Results of the Stability Analysis and Discussion  
The aforementioned bifurcation diagrams can be utilized to make conjectures regarding the stability of Type 1 symmetric and asymmetric solitons [32; 57]. According to the elementary bifurcation theory, for those values of \(\lambda\) for which the asymmetric and symmetric solitons exist together (i.e., when \(\lambda\) is in the range \(|\lambda|<\lambda_{c}\)), the symmetric solitons (i.e., solitons corresponding to the dotted lines in Figure 4) should be unstable and the asymmetric ones should be stable. When \(|\lambda|>\lambda_{c}\), there are only symmetric solitons in which case the bifurcation theory predicts that the symmetric solitons would be stable. Since bifurcation diagrams cannot be constructed for Type 2 solitons (see Section 3), their stability must be determined numerically. To determine the stable regions within the bandgap and to check the validity of the conclusions of the bifurcation theory, we have performed a systematic and extensive numerical stability analysis of the symmetric and asymmetric Type 1 and Type 2 solitons. To this end, a symmetrized split-step Fourier scheme has been employed (see Reference [58]) to solve Equation (1). In addition, to prevent reflection of radiation, absorbing boundary conditions have been incorporated into our simulations. The numerically-obtained solutions were used as the initial configuration for the numerical simulations. The evolution of solitons was simulated up to \(t=2000\). The noise inherent in the numerical scheme was found to be adequate for the onset of the instability development. The results of our numerical simulations fully corroborate the predictions of the bifurcation theory. As is shown in Figure 5, for the parameters where the Type 1 symmetric and asymmetric GSs exist together, the asymmetric soliton is stable (Figure 5a) and the symmetric one is unstable (Figure 5b,c). Additionally, Figure 5b,c demonstrate that the evolution of the unstable Type 1 symmetric solitons is accompanied by energy loss as radiation and eventually they evolve to a Type 1 asymmetric soliton. As regards the stability of Type 2 gap solitons, the simulations show that there is a region within the bandgap where Type 2 asymmetric solitons are stable. An example of the stable Type 2 asymmetric soliton is shown in Figure 6a. Type 2 symmetric solitons, on the other hand, are found to be always unstable. An example of the evolution of a Type 2 symmetric soliton is displayed in Figure 6b,c. It is noteworthy that the symmetric Type 2 soliton evolves to an asymmetric _moving_ Type 2 soliton. This confirms that stable moving Type 2 asymmetric solitons are supported by Equation (1). Figure 5: Propagation of Type 1 asymmetric an symmetric solitons with \(\omega=0.0\), \(\eta=0.15\), \(\lambda=0.1\), and \(m=0.1\). (**a**) stable asymmetric soliton (\(u_{1}-\)component), (**b**) unstable symmetric soliton (\(u_{1}-\)component) and (**c**) unstable symmetric soliton (\(u_{2}-\)component). Figure 6: Propagation of Type 2 asymmetric and symmetric solitons with \(\omega=-0.5\), \(\eta=0.9\), \(\lambda=0.1\), and \(m=0.1\). (**a**) stable asymmetric Type 2 soliton (\(u_{1}-\)component), (**b**) unstable symmetric soliton (\(u_{1}-\)component) and (**c**) unstable symmetric soliton (\(u_{2}-\)component). The propagation of unstable Type 1 and Type 2 asymmetric solitons might result in different outcomes. For example, as is shown in Figure 6(a), the asymmetric unstable Type 1 soliton sheds some of its energy during the propagation and then rearranges itself into a Type 1 asymmetric moving stable soliton. On the other hand, the evolution of Type 2 asymmetric solitons might result in either small amount of radiation followed by the formation of another quiescent stable Type 2 asymmetric soliton (Figure 6(b)) or complete destruction of the soliton (Figure 6(c)). One of the goals of the stability analysis is to identify the stable regions inside the bandgap for both types of solitons.