Hamiltonian analysis and dynamical behavior of bright, dark and other multiple soliton solutions of the Katugampola-fractional reduced spin Hirota-Maxwell-Bloch system

In this research, we investigated the integrable fractional reduced spin Hirota-Maxwell-Bloch system with Katugampola fractional derivatives, a crucial model for analyzing the femtosecond pulses transmitted within an erbium-doped fiber. The modified unified method and the $(G^{\prime }/G)$-expansion techniques were employed to acquire analytical soliton solutions comprising bright soliton, mix dark-bright soliton, dark soliton, perturbed dark and bright soliton, multi-soliton, and periodic dark-bright soliton. The fundamental mechanics of the model was revealed by dynamically displaying some of the found solutions using 2D and 3D graphs. The $\rho$-derivative framework was used to analyze the effect of the space-fractional derivative on the supplied model, offering a more dynamic and applicable way to improve the accuracy of the findings. Additionally, a range of graphical representations were used to show the perturbed system's time series plots and the planar system's phase portraits under the Hamiltonian analysis to emphasize the model's significance and dynamic behavior in erbium-doped fiber. Our findings of this study are expected to have important ramifications for soliton theory, erbium-doped fiber, optical fibers, physical engineering, and nonlinear dynamics. Additionally, the study shows that the $(\frac{G^{\prime }}{G})$-expansion method and the modified unified approach are simple, robust, and efficient methods that produce many soliton solutions for a range of nonlinear fractional partial differential equations in the mathematical sciences.