## 4 Analysis of the Modulational Instability
### The Coupler with Opposite Signs of the Dispersion in the Two Cores  
The case of the opposite (\"mixed\") GVD signs in the two cores of the coupler, which corresponds to \(\alpha<0\) in Equation (5), is obviously interesting, as well [47]. For this purpose, we assume the anomalous and normal GVD in the bar and cross channels, respectively.  
Figure 9 shows the effect of the CW power, \(P\), on the MI in the mixed-GVD coupler. The figure demonstrates that the MI gain and bandwidth monotonically increase with the growth of \(P\). It should be noted that the spectra obtained for this case are somewhat different in comparison with the conventional MI spectra, as the gain is stretched over a broad interval of the perturbation frequency when the CW power is low (\(P<1\)). In the present case, the effect of the coupling coefficient, \(\kappa\), on the MI, which is shown in Figure 10, is essentially the same as demonstrated above for the coupler with the normal GVD in both cores; see Figure 6a. Namely, the MI gain and bandwidth increase with the growth of \(\kappa\).  
The effects of the negative value of the ratio of the GVD coefficients, \(\alpha<0\), and the ratio of the nonlinearity coefficients (\(\Gamma\)) in the two cores are shown in Figure 11. Similar to the coupler with the anomalous GVD in each core, cf. Figure 3b, the increase of \(\alpha\) (see Figure 11a) leads to shrinkage of the MI band. Like in the coupler with the anomalous GVD in both cores, cf. Figure 3c, the MI gain increases with the growth of \(\Gamma\), which is depicted in Figure 11b; however, the difference is that, in the present case of the mixed-GVD coupler, the bandwidth is not affected by the variation of \(\Gamma\).  
Figure 12 displays quite nontrivial evolution of the MI spectra with the variation of the group-velocity mismatch (walk-off) between the cores, \(\rho\) in Equation (5). The evolution is very different from what is demonstrated above for the coupler with the anomalous GVD in both cores, cf. Figure 4a. Namely, Figure 12 shows that the increase of \(\rho\) from zero to one suppresses the MI, which completely vanishes at \(\rho=1\). The system recovers the MI, which features monotonically increasing gain and bandwidth, with the further increase of \(\rho\) to values \(\rho>1\).  
Figure 8: The MI gain spectra in the normal-GVD regime (\(\sigma_{  
Next, we consider the impact of the asymmetry parameter \(\eta\) in the CW state; see Equation (8). As show in Figure 13, there is no MI at small values of \(\eta\), such as \(\eta=0.1\). With the subsequent increase of \(\eta\) up to \(\eta=1\), the MI gain and bandwidth increase, similar to what was observed above in the coupler with anomalous GVD in both cores; see Figure 4b. However, the situation becomes completely different at \(\eta>1\), when the CW amplitude is higher in the bar channel: the MI band splits into two narrower ones, with smaller values of the gain.  
Figure 11: The MI gain spectra in the mixed-GVD coupler (\(\alpha<0\)). (**a**) The results for different negative values of the ratio of the GVD coefficients in the two cores (\(\alpha\)), which are indicated near the curves for \(P=2,\eta=0.5,\kappa=1.1,\Gamma=2\) with \(\rho=0.01\). (**b**) The results for different values of the ratio of the nonlinearity coefficients in the two cores (\(\Gamma\)), which are indicated near the curves. Parameters are same as in (a), except for \(\kappa=1\) and \(\alpha=-0.1\).  
Figure 12: The MI gain spectra for different values of the group-velocity mismatch (walk-off) between the cores in the mixed-GVD coupler (\(\alpha<0\)) for \(P=\kappa=1,\eta=0.5,\alpha=-0.1,\Gamma=2\).  
Figure 10: The MI gain spectra for different values of the coupling coefficient, \(\kappa\) (indicated near the curves), in the mixed-GVD coupler (\(\alpha<0\)) for \(P=1,\eta=0.7,\alpha=-0.1,\Gamma=2\) with \(\rho=0.01\).