Bifurcation, Chaotic, Sensitivity Analysis, and Optical Soliton Profiles for the Spin Hirota–Maxwell–Bloch Equation in an Erbium-Doped Fiber

In this work, the reduced spin Hirota–Maxwell–Bloch (rsHMB) equation is examined analytically. This model is useful for characterizing the propagation of femtosecond pulses in an erbium-doped fiber. The optical soliton solutions of the introduced rsHMB problem are constructed using the Jacobi elliptic function (JEF) expansion technique. This is efficient technique for the exact solitary wave solution of nonlinear partial differential equations (NLPDEs). The multiple soliton profiles are observed in the single and combined wave solutions, such as, bright soliton, dark soliton, singular soliton, kink soliton, and mixed solitons. All of the recently produced soliton solutions are verified by returning them to the relevant system using Wolfram Mathematica soft computation. Additionally, the analysis of bifurcation is investigated and the problem is transformed into a planer dynamical system utilizing a specific transformation. Additionally, by adding definite periodic pressures to the model under consideration, the quasiperiodic solution for the perturbed system is examined numerically. Two-dimensional and three-dimensional phase portraits are plotted in relation to the perturbed models parameter.