Dynamical structures of optical solitons for highly dispersive perturbed NLSE with \(\beta \)-fractional derivatives and a sextic power-law refractive index using a novel approach

In this paper, we investigate the highly dispersive perturbed nonlinear Schrödinger equation (NLSE) with \(\beta \)-fractional derivatives, generalized nonlocal laws and sextic-power law refractive index. This equation is crucial for modeling complex phenomena in nonlinear optics, such as soliton formation, light pulse propagation in optical fibers, and light wave control, with potential applications in designing efficient optical communication devices. Furthermore, it provides a framework for understanding the intricate interactions between high dispersion, nonlocality, and complex nonlinearity, contributing to the development of new theories in wave physics. To accomplish this, we use the modified extended direct algebraic method. A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, combo bright-dark and singular soliton solutions. Additionally, singular periodic solutions, rational and exponential solutions. Furthermore, graphical simulations are presented that highlight the distinctive characteristics of these solutions. Compared to Nofal et al. (Optik 228:166120, 2021), the proposed technique produced novel and diversified results. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. The extracted solutions confirmed the efficacy and strength of the current approach. The parameter constraints ensure the existence of the obtained soliton solutions. It is worth noting that the proposed method, being effective, consistent, and influential, can be applied to solve various other physical models and related disciplines.