## 7 Discussions and Conclusions  
It was known that a multi-component Manakov system can effectively model wave propagation in a multicore optical fiber [22]. Besides temporal waveguides, such Manakov systems are relevant in other settings in optical physics too. For spatial solitons, diffraction will play the role of group velocity dispersion, and continuous variations of diffraction, nonlinearity, and gain/loss might lead to novel rogue wave patterns [47]. Similarly, Manakov soliton can arise for biased guest-host photorefractive polymer too [48]. Furthermore, reductive perturbation techniques can be employed to establish Manakov equations as approximations for propagation of electromagnetic fields along isotropic chiral metamaterials [49]. In this work, a theoretical study is performed to understand the increasing complexity of the Manakov models with larger number of components. It will be worthwhile if analytical and computational predictions here can be verified in experiments in the future. More precisely, rogue wave solutions for the defocusing three-component Manakov system with group velocity mismatch are derived by the Hirota bilinear method. The nonlinear coupling effect is highlighted, namely, the extension of existence regime of rogue waves and the enhancement of amplitude. The main focus of the work is to demonstrate the generation of rogue wave from baseband disturbance. Recently, the onset of baseband modulation instability has been proven to be equivalent to the existence condition of rogue waves in several systems. In this work, the role of low frequency disturbance in the formation of rogue waves is further consolidated through numerical simulations. By perturbing the plane wave solution by a single mode disturbance with a small frequency, rogue wave modes with configuration similar to the analytical rogue wave solutions can be generated. This idea can be generalized to approximate rogue wave modes in non-integrable systems where most analytical methods fail. The proposed numerical methodology can greatly enrich our knowledge of rogue waves in such systems. Detailed investigations on evolution of rogue waves in general systems of coupled nonlinear Schrodinger equations without any restriction on coefficients would be conducted in the future. Moreover, modified NLS equation or the corresponding systems can be studied [50]. Secondly, the single mode wavy disturbance to a plane wave can generate breathers as well. It is well-known that the formation of breathers is closely related to MI. Depending on the MI spectrum, both symmetric and asymmetric breathers can be generated from a single mode wavy disturbance. For the case where there are multiple unstable bands, the unstable mode can generate multiple breathers which superimpose to form an asymmetric breather. On the other hand, a conventional symmetric breather is generated if there exists only one unstable band. In conclusions, rogue wave and breather formation are closely related to the nature of the MI spectrum. Through the study of the MI spectrum supplemented with numerical simulations, more intriguing wave dynamics of general non-integrable systems can be revealed in the future. Theoretically, three-component or multi-component systems with variable coefficients and external potential can be further investigated through a similarity transformation [13]. Recently, nonlocal equations have been widely studied due to their _PT_ symmetric property [51]. Many nonlocal evolution equations display remarkable similarities in comparison with intensively studied classical ones, e.g., the nonlocal NLS equation  
\[iA_{z}+A_{tt}+AA^{*}(-t,z)A=0, \tag{9}\]  
which can be analyzed by direct and inverse scattering techniques [51]. Indeed, the nonlocal NLS model (Equation (9)) also possesses an infinite number of conservation laws.