Solitons unveilings and modulation instability analysis for sixth-order coupled nonlinear Schrödinger equations in fiber bragg gratings

This work investigated analytical solutions for a coupled system of nonlinear perturbed Schrödinger equations in fiber Bragg gratings (FBGs), characterized by sixth-order dispersion and a combination of Kerr and parabolic nonlocal nonlinear refractive indices. Chromatic dispersion, which restricts wave propagation in standard optical fibers, was effectively compensated using FBGs, making them indispensable in modern optical networks. In this study, the modified Sardar sub-equation technique (MSSE) was applied to the system for the first time. This method was chosen for its advantages, including low computational cost, high consistency, and simplicity in calculations. The novelty of this work lied in the derivation of new analytical solutions, such as exponential, singular periodic, hyperbolic, and rational solutions, which have not been previously reported in the literature. Additionally, bright gap solitons and singular gap solitons, previously studied, were also obtained. All solutions were rigorously verified by direct substitution into the system. Another significant contribution of this work was the derivation of modulation instability (MI) analysis using linear stability analysis. For the first time in the literature, an analytical expression for the MI gain spectrum was derived. This gain spectrum depended on key parameters such as normalized power, perturbation wave number, dispersion coefficients, phase modulation coefficients, and nonlinearity coefficients. The study also included 2D and 3D graphical representations of selected exact solutions, with parameters chosen to satisfy specific limiting conditions, as well as visual illustrations of the MI gain spectrum. The solutions derived in this work have profound implications for optical communication systems. Exponential and hyperbolic solutions can model pulse propagation in FBGs with high accuracy, enabling better design of dispersion-compensating devices and improving signal integrity over long distances. Singular periodic and rational solutions provided insights into the behavior of nonlinear waves in FBGs, which can be exploited for advanced signal processing applications, such as pulse shaping and wavelength conversion. Bright and singular-gap solitons were crucial for maintaining stable signal transmission in FBG-based systems, particularly in high-power scenarios where nonlinear effects were significant. The MI analysis further enhanced the practical relevance of this work. By understanding the conditions under which MI occured, engineers can design FBG systems that minimize signal degradation and optimize performance. The MI gain spectrum provided a tool to predict and control instability, ensuring robust and efficient optical communication networks. This work not only advanced the theoretical understanding of nonlinear wave dynamics in FBGs but also offered practical tools and solutions for improving optical communication systems. The derived solutions and MI analysis have direct applications in enhancing signal stability, dispersion management, and overall system performance, making this research highly relevant to the field of photonics and optical engineering.