Analytical Exploration of Optical Structures in an Integrable Reduced Spin Hirota–Maxwell–Bloch System

An integrable reduced spin Hirota–Maxwell–Bloch (rsHMB) equation is a crucial model for describing femtosecond pulse propagation in erbium-doped fiber, soliton dynamics, and magnetization reversal. The main objective of this research is to derive the exact solution of the reduced integrable spin Hirota–Maxwell–Bloch (rsHMB) equation, as the solutions offer mathematical tools for understanding wave behaviour and have different practical applications. In this research, we use two different analytical methods, the Kumar Malik method, and the modified Sardar sub-equation method, to find the analytical solutions of the equation. The equation is simplified by converting it to a nonlinear ordinary differential equation by the use of a travelling wave transformation. By using these methods, we have obtained different type of solutions to the rsHMB equation, such as Jacobi elliptic functions, exponential, hyperbolic, and trigonometric functions that describe the properties of the model. These analytical solutions are further represent in 3D, 2D and density graphs that show the behaviour of different soliton solutions. We have specifically examined how multiple parameters affect the wave width, velocity, and other crucial characteristics. The advantages of Kumar Malik and Sardar sub-equation method are that these provide many kinds of solitons, such as dark, bright, singular, periodic singular, combined dark–singular and combined dark–bright, singular kink waves, bright solitons, and breather waves. It is important to note that the recommended methods for solving nonlinear partial differential equation are competent, credible, and intriguing tools for analysis. Furthermore, the results are a useful tool for understanding the complicated behaviour of physical systems and providing inspiration for further study.