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Modulation instability and rogue wave structures have been investigated in this work. This study is an extension of the work in Abbagari (2023), where a nonlinear Schrödinger equation with higher-order dispersion is derived to show only the development of the modulated waves bounded to bright soliton as a nonlinear exhibition of modulation instability. Here, the coupled nonlinear Schrödinger equation is derived by using the multi-scale scheme. An overview of the analytical calculations of the perturbed plane wave is carried out to show the effects of the nonlinear chain parameters on the modulation instability growth rate and bandwidths. The interest of this study lies equally in the nonlinear modes of excitation, where solitonic waves are generated under certain conditions in lower and upper frequency bands. On the other hand, relevant results have been developed to show the features of the type I and type II rogue waves of the Manakov system. Such investigations are obtained under the variation of the interaction potential parameters and the free parameter of the similarity method. Via a numerical simulation, rogue wave structures have been generated as a consequence of the long-time evolution of the perturbed plane wave. At a specific time of propagation, another localized object has been obtained to show the Akhmediev breathers and Kuznetsov-Ma solitons clusters under a strong perturbed wave number. These results have opened up new features, and many applications could follow in the future. • Modulation instability (MI) and rogue wave (RW) have been investigated in this work. • The CNLSE is derived by using the multi-scale scheme. • The effects of chain parameters on the MI growth rate and bandwidths have been shown. • The features of the type I and type II RWs of the Manakov system have been show. • Via a numerical simulation, RWs structures have been generated. [ABSTRACT FROM AUTHOR]

This work investigates the modulation instability and wave patterns in the extended nonlinear Schrödinger equation with higher-order dispersion and cubic–quintic nonlinearity. We use the one-dimensional monoatomic chain with anharmonic potential to derive the discrete model. From the multiple scale method combined with a quasidiscreteness approximation, we derive the cubic–quintic nonlinear schrödinger equation, and thereafter an expression of the modulation instability gain is obtained by using a linearized expression. One notices that the modulation instability growth is sensitive to the higher-order dispersion and nonlinearities terms. A split-step Fourier method is used to assess the analytical predictions. A long evolution of the continuous wave is shown to lead to the formation of the bright soliton, and Akhemediev breathers also emerge to manifest the modulation instability. We have also demonstrated that the excitation wave number generates the train of waves to confirm the fact that the continuous wave can grow exponentially with any value of the latter. We mention equally that the model of the extended cubic–quintic nonlinearity with complex envelope has opened new features of the modulated wave patterns in monoatomic chains. • The MI and waves patterns of the NLSE are investigated. • To obtain CQ NLSE we use multiple scale method and quasi-discreteness approximation. • A split-step Fourier method is used to assess the analytical predictions. • Bright solitons and Akhemediev breathers also emerges to manifestthe MI. • The CW can grow exponentially with any value of the excitation wave number. [ABSTRACT FROM AUTHOR]

In this work, we studied the behavior of the modulation instability (MI) growth rate and modulated wave (MW) bright and dark-soliton in nonlinear optical fibers (OFs) where both birefringence and four-wave mixing (4WM) terms are employed. We have shown the effects of the 4WM and the group velocity dispersion (GVD) on the MI gain and MI bands as well as on the MWs patterns. For certain conditions on the coupled nonlinear Schrödinger equation (CNLSE) parameters, we exhibited stable and unstable zones of the MI. More precisely, in a normal dispersion regime, we shown how the variation of 4WM can generate additional sides lobes and consequently the formation of new MI bands. Furthermore, it is observed that when the coupling coefficient (CC) increases, its effects could reduce the stability zones by increasing strongly the number of sides lobes in a normal dispersion regime. To seek the behavior of the MWs patterns, we used numerical investigation (NI). From these results, it has been observed the propagation of the MW bright-soliton and dark-soliton with the effects of the 4WM. We noticed that the MWs patterns can preserve high energy on the peak and keep constant its shapes in birefringent OFs. These results show that the 4WM and the GVD behave as being energy sources for MI and MWs. Our finding can help to manufacture OF tools and could open the way for the use of the 4WM and cross-phase modulation (XPM) to generate MI bands and long-live temporal MWs patterns. • Modulation Instability. • Modulated wave pattern. • Cross-Phase Modulation. • Solitons. [ABSTRACT FROM AUTHOR]