Exploring nonlinear dynamics and soliton structures in the spin reduced Hirota-Maxwell-Bloch system via Atangana’s conformable operator

Abstract

In this work, we analyze the integrable reduced spin Hirota-Maxwell-Bloch (rsHMB) equation, which is an essential model for femtosecond pulse propagation in vector-soliton interactions, magnetization-reversal phenomena, and erbium-doped fibers, along with Atangana’s conformable derivative, thereby incorporating memory effects into the spin-field coupling. Two analytic techniques are used to construct exact, closed-form solutions: (i) the complete discriminant system method (CDSM) of polynomials, which produces dark, bright, and mixed soliton families together with periodic wave packets, and (ii) a generalized Arnous method that yields novel multi-peak and W-shaped solitary patterns. A subsequent bifurcation analysis reveals a rich hierarchy of dynamical regimes; specifically, elastic-medium stiffness and rippling amplitude emerge as the primary control parameters driving nonlinear free vibration. To chart the transition from order to chaos, we employ a comprehensive set of nonlinear-dynamics diagnostics. Time-series traces, phase portraits, multistability tests, Poincaré sections, and strange-attractor reconstructions expose multiple coexisting attractors and sensitive dependence on distinct initial conditions. Quantitatively, a Lyapunov-exponent spectrum verifies the existence of at least one positive exponent, while power-spectrum densities and return maps exhibit broadband signatures consistent with deterministic chaos. A dedicated sensitivity analysis further demonstrates the butterfly effect, showing that infinitesimal perturbations precipitate markedly various long-term results. These results expand the solution space of the rsHMB system in the context of conformable fractional calculus, identify key mechanical parameters influencing its nonlinear oscillatory structure, and offer clear experimental indicators like bifurcation thresholds, chaotic bandwidths, and fractal attractor patterns for the control and detection of ultrafast spin-photon dynamics in magnetic media and optical fiber applications. The study thus provides both theoretical insight and practical guidelines for the development of high-speed, soliton-based spintronic and photonic devices.