## 4 Analysis of the Modulational Instability
### The Normal-Dispersion Regime  
The combination of the self-focusing Kerr nonlinearity and normal GVD usually supports stable CWs. However, as mentioned in the Introduction, MI may occur under the normal GVD in more complex systems, including couplers. Following the pattern of the MI investigation presented above for the anomalous GVD, we first consider the effect of the CW power, \(P\), on the MI gain. We also compare the instability spectrum of the asymmetric system with that of the symmetric one in Figure 5a,b, respectively. As seen in Figure 5, in both cases, two distinct MI bands determine the instability, and (similar to the anomalous-GVD regime) the MI gain of the asymmetric system linearly grows with \(P\), featuring a broad bandwidth. To illustrate the essential effect of the coupling coefficient \(\kappa\), Figure 6a depicts the MI gain spectra for various values of \(\kappa\). Naturally, no MI takes place in the normal-GVD regime in the absence of the coupling, \(\kappa=0\). It is worthy to note the appearance of two separated MI bands at \(\kappa>1\), the MI gain increasing in both bands, along with their widths, with the growth of \(\kappa\). The effect of the relative difference in the magnitude of the normal GVD in the two cores, \(\alpha\), is shown in Figure 6b. Like in the anomalous-GVD regime, here, as well, the MI bandwidth is infinite for \(\alpha=0\) (it also contains a separate finite MI band). The MI spectrum features two separate bands at \(\alpha>0\) and the largest gain at \(\alpha=0.1\). The gain decreases with the subsequent increase of \(\alpha\). Figure 7a shows the effect of the relative difference in the effective mode areas between the two channels, i.e., the ratio of the nonlinearity coefficient, \(\Gamma\). It is seen that no MI occurs when the cross channel is linear (\(\Gamma=0\)), and two distinct MI bands emerge and expand, featuring a growing largest value of the instability gain, with the increase of \(\Gamma\). The influence of the group-velocity mismatch (walk-off between the cores), \(\rho\), is depicted in Figure 7b. Once again, the MI appears in the form of two separated bands. The MI gain and bandwidth nontrivially depend on \(\rho\): at \(\rho<1\) the low-frequency band is narrower, with smaller values of the instability gain, while at \(\rho\geq 1\), the situation is inverted. We have also analyzed the effect of the CW's asymmetry \(\eta\) (see Equation (8)) on the MI in the normal-GVD regime. No MI occurs for small values of \(\eta\), viz. \(\eta<0.2\). At \(\eta>0.2\) (in particular, at \(\eta=0.5\)), there again emerge two separate MI bands, as shown in Figure 8. The MI gain and  
Figure 4: The MI gain spectra in the anomalous-GVD regime (\(\sigma_{1}=1\)). (**a**) Results for different values of the group-velocity mismatch (\(\rho\) in Equation (5)), which are indicated near the curves. Other parameters are \(P=\kappa=\Gamma=1\), \(\eta=0.5\) and \(\alpha=0.1\) (**b**) Results for different values of asymmetry ratio \(\eta\) of the CW state (see Equation (8)), which are indicated near the curves. Other parameters are \(P=0.5,\kappa=1\), \(\alpha=0.1,\Gamma=2\) and \(\rho=0.01\). bandwidth attain their maxima at \(\eta=1\) (equal amplitudes of the CW in the two cores), decreasing with the further increase of \(\eta\). Figure 5: Contour plots showing the dependence of the MI gain on the CW power, \(P\), in the normal-GVD regime (\(\sigma=-1\)) in the symmetric and asymmetric systems. Parameters of the symmetric system (**a**) are \(\alpha=\Gamma=1\), \(\eta=2\), \(\kappa=0.9\) and \(\rho=\chi=0\). For the asymmetric system (**b**), \(\eta=0.5,\alpha=0.1,\Gamma=2,\kappa=1.1\) and \(\rho=0.01\), \(\chi=-0.825+0.7P\). Figure 6: **(a)** The MI gain spectra in the normal-GVD regime (\(\sigma_{1}=-1\)) for different values of (**a**) the coupling coefficient, \(\kappa\), indicated near the curves. Other parameters are \(P=1,\eta=0.5\), \(\alpha=0.1\), \(\Gamma=2\) and \(\rho=0.01\).