## 1 Introduction  
The modulational instability (MI) is a ubiquitous phenomenon originating from the interplay of linear dispersion or diffraction and the nonlinear self-interaction of wave fields. This effect was first theoretically identified by [PERSON] and [PERSON] in 1967 for waves on deep water [1]; hence, MI is often called the Benjamin-Feir instability. Studies of the MI draw steadily growing interest in nonlinear optics [2; 3; 4], fluid dynamics [5; 6], Bose-Einstein condensates [7; 8; 9], plasma physics [10; 11] and other fields. In its standard form, the MI applies to continuous waves (CWs) or quasi-CW states in media featuring cubic (Kerr) self-focusing nonlinearity and anomalous group-velocity dispersion (GVD), giving rise to the instability against infinitesimal perturbations in the form of amplitude and phase modulations, which eventually generates trains of soliton-like pulses [12]. MI can also be observed in the normal-GVD regime in systems incorporating additional ingredients, such as the cross-phase modulation interaction between two components [13], in the case of the co-propagation of optical fields and other effects, in particular the loss dispersion [14] or fourth-order GVD [15]. In all of these cases, destabilizing perturbation may originate from quantum noise or from an additional weak frequency-shifted wave [16]. Based on the nature of the underlying optical propagation, the MI is classified as the temporal (longitudinal) instability [17; 18], if the CW is subject to the GVD in fibers, or spatial (transverse) instability [19], if the CW state experiences the action of diffraction in a planar waveguide. More general spatio-temporal MI occurs in bulk optical media when both the GVD and diffraction are essential [20]. The MI has found many important applications, including the creation of pulses with ultra-high repetition rates [21; 22], the expansion of the bandwidth of Raman fiber amplifiers [23], the generation of optical supercontinuum [24] and all-optical switching [25]. In the context of nonlinear fiber optics, MI can also drive the four-wave mixing initiated by the interaction of a signal wave with random noise [13]. MI is also often regarded as a precursor to soliton formation, since the same nonlinear Schrodinger equation, which governs the MI, gives rise to stable solitary pulses. Indeed, the breakup of the original CW into soliton arrays may be an eventual outcome of the development of the MI [16]. Starting from the theoretical analysis by [PERSON] [26], followed by the experimental verification [27], nonlinear directional couplers (NLDC), which are built as dual-core fibers, have been one of the promising elements of integrated photonic circuits for the realization of ultrafast all-optical switches, as well as a subject of intensive fundamental studies [25; 28; 29; 30; 31; 32; 33]. The operation of the NLDC is governed by the interplay of the Kerr self-focusing, which induces a change in the refractive index in each core, intra-core linear GVD, and linear coupling between the cores. The linear-coupling coefficient determines the critical value of the power, which gives rise to the spontaneous breaking of the symmetry between the two cores [34]. Based on such power-dependent transmission characteristics, many applications of the NLDC have been proposed, such as all-optical switching and power splitting [25], logic operations [35; 36], pulse compression [37] and bistability [38]. The MI dynamics in NLDC models was investigated in many works. In particular, in [39], [PERSON] et al., who first studied soliton switching in NLDC [25], also investigated the MI, considering different combinations of linear and nonlinear effects in a saturable nonlinear medium. In [40], the MI was investigated for antisymmetric and asymmetric CW states in the dual-core fibers, demonstrating that they are subject to the MI even in the normal-GVD regime.