## 5 Direct Simulations  
The analytical results obtained above for the MI have been checked against numerical calculations of the instability spectra. Numerical methods are actually more relevant for direct simulations of the nonlinear evolution of the MI, which was analyzed above in the linear approximation. The simulations were carried out by dint of the well-known split-step Fourier method [48] (using MATLAB). Most results displayed below were obtained using numerical meshes with 512 Fourier points and periodic boundary conditions with respect to variable \(\tau\). Simulations performed with denser meshes have produced virtually identical results. Furthermore, results of the nonlinear development of the MI are not sensitive to details of initial small perturbations, which initiate the onset of the MI. The initial conditions were taken in the form of the CW to which a small periodic perturbation was added:  
\[u_{j}(0,\tau)=A_{j}+a_{0}\cos(\omega_{0}\tau),\quad(j=1,2), \tag{19}\]  
where \(a_{0}\) is a small amplitude of the perturbation and \(\omega_{0}\) is its frequency. Various outcomes of the MI development for CW states with different parameters are displayed in Figures 14-23. First, in Figure 14, we show the results for the symmetric coupler in the anomalous-GVD regime when the amplitudes of two CW components are equal (\(A_{1}=A_{2}=1\)). As seen in the figure, a periodic chain of well-shaped soliton-like pulses is produced on top of the nonzero background in both cores. Longer simulations demonstrate regular dynamics of the quasi-soliton arrays. In this work, we do not aim to study the latter in detail, as it is not closely related to the initial MI. Figure 14: The evolution of the MI in the symmetric coupler with anomalous GVD (\(\sigma=1\)) in the bar (**a**) and cross (**b**) channels for equal amplitudes of the underlying CW state, \(A_{1}=A_{2}=1\), with perturbation parameters \(a_{0}=0.0001\) and \(\omega_{0}=1\), in Equation (19). Other parameters are \(\alpha=\Gamma=\kappa=1\) and \(\rho=\chi=0\). Figure 13: The MI gain spectra for different values of the asymmetry ratio, \(\eta\), of the CW state (see Equation (8)), which are indicated near the figures, in the mixed-GVD coupler (\(\alpha<0\)) for \(P=0.5,\kappa=1.1,\alpha=-0.1,\Gamma=2\) with \(\rho=0.01\). We now turn to simulations of the MI in the asymmetric coupler and the analysis of effects of its different parameters. The impact of the group-velocity mismatch (walk-off) between the cores in the anomalous-GVD regime is presented in Figure 15. As seen in the figure, pulses generated by the MI drift away from their original positions, which implies spontaneous symmetry breaking, as a particular drift direction is selected by the system. We have also investigated the spectral evolution of the MI for different values of the group-velocity mismatch. The results (not shown here in detail) corroborate, in particular, that the group-velocity mismatch has no impact on the instability spectrum, as predicted by the analytical results in Figure 4a. Further, Figure 16 shows the influence of the phase-velocity mismatch on the MI evolution in the anomalous-GVD regime. In this case, the main effects are oscillations of the background and retaining of the power chiefly in the bar channel. The role of the ratio of the GVD coefficients in the two cores, \(\alpha\), is shown in Figures 17 and 18, for the case of the anomalous GVD in both cores. In the case of zero GVD in the cross channel (\(\alpha=0\)) (Figure 17) shows that a chain of quasi-solitons with growing amplitudes is generated on top of a nonzero background in the bar channel, while narrow growing peaks emerge at edges of the background in the cross channel. If \(\alpha\) increases to \(\alpha=2\), the former picture is essentially reversed, so that a chain of solitons on top of the background appears in the cross channel, and a chain of very narrow solitons is generated in the bar channel. In all of these cases, the soliton chains keep the initial modulation period, \(2\pi/\omega_{0}\).