## 6 Conclusions  
In this work, we have investigated the MI (modulational instability) in the model of asymmetric dual-core NLDCs (nonlinear directional couplers), based on the system of nonlinear Schrodinger equations, which include differences in the GVD and nonlinearity coefficient in the two cores, as well as the group- and phase-velocity mismatch between them. The MI of symmetric and asymmetric CW states in the NLDC against small perturbations was investigated using the linearized equations for the perturbations. This was followed by direct simulations to investigate the nonlinear development of the MI. First, we have considered the dependence of the MI gain spectra on the total power of the two-component CW states in the coupler with the anomalous sign of the GVD in both cores. It was found that the MI bands in the asymmetric couplers are broader in comparison with their symmetric counterparts. Then, we focused on the impact of the magnitude of the inter-core coupling coefficient, \(\kappa\), demonstrating that the increase of \(\kappa\) leads to gradual suppression of the MI. Next, a large GVD coefficient in the bar channel, in comparison with the cross channel, generates very broad MI spectra with large values of the instability gain. If the asymmetry between the cores is introduced only through the difference in the GVD coefficients, high values of the group-velocity mismatch cause splitting of  
Figure 23: Creation of a chaotic (turbulent) state by the MI in the bar (**a**) and cross (**b**) channels in the normal-GVD dispersion regime (\(\sigma=-1\)) for CW amplitudes \(A_{1}=2,A_{2}=0.001\), with perturbation parameters \(a_{0}=0.09,\omega_{0}=1\). Other parameters are \(\alpha=2,\Gamma=0.5,\rho=0.01,\chi=0.02\) and \(\kappa=1\). Figure 22: The MI evolution in the bar (**a**) and cross (**b**) channels in the normal-GVD regime (\(\sigma=-1\)) for the amplitudes of the CW components \(A_{1}=A_{2}=1\) and perturbation parameters \(a_{0}=0.002,\omega_{0}=1\). Other parameters are \(\alpha=2,\Gamma=1,\rho=0.01,\chi=0.001\) and \(\kappa=1\). the single MI band into two. The effect of asymmetry between two components of the CW state, \(\eta\), was identified as well. It was found that the MI gain and bandwidth reduce with the increase of \(\eta\) from small values to one, while the further increase of \(\eta\) leads to shrinkage of the MI band. Next, the MI was explored in the normal-GVD regime, in which the MI occurs in two separated spectral bands. The increase of the coupling coefficient makes the size of the MI gain in the two bands strongly different. The influence of the difference in the GVD and nonlinearity coefficients was analyzed, as well. The increase of these coefficients leads, respectively, to the decrease and increase of the MI gain in the two bands. Noteworthy results were produced by the analysis of the MI in the coupler with opposite signs of the GVD in the cores. While the difference in the negative values of the GVD coefficient, and in the nonlinearity coefficients, produce approximately the same effects as in the anomalous-GVD regime, the response to the increase of the coupling coefficient is similar to that in the case of the normal GVD, leading to the increase of the MI gain. A notable effect was observed with the variation of the group-velocity mismatch, \(\rho\), between the cores: the increase of \(\rho\) from small values to one suppresses the MI, which disappears at \(\rho=1\). It appears again and enhances at \(\rho>1\). The asymmetry ratio of the two components of the underlying CW state, \(\eta\), also produces a nontrivial effect: while the MI is absent at small values of \(\eta\), it appears at \(\eta\gtrsim 0.5\) in the form of a single spectral band, which grows up to \(\eta=1\) and then splits into two bands. Finally, we have also performed systematic simulation of the nonlinear development of the MI in different regimes, which were studied analytically. Typical outcomes feature the generation of periodic chains of growing peaks in the anomalous-GVD regime.