Bright and dark optical solutions in birefringent fibers with polynomial law of non-linear refractive index via semi-analytical technique Shehu HPM

This study addresses the propagation and stability of bright and dark optical soliton solutions in birefringent fibers governed by a nonlinear refractive index modeled through a polynomial law. Understanding such nonlinear wave dynamics is crucial for enhancing high-speed optical communication systems. To address this problem, a semi-analytical approach (Shehu Homotopy Perturbation Method) is proposed which combines strengths of Shehu transform and the classical Homotopy Perturbation Method. This fusion enables derivation of approximate analytical solutions without any need for discretization, linearization, or quasi-linearization, thereby eliminating associated numerical errors. The effectiveness of proposed method is demonstrated through a detailed comparison between exact and approximate solutions at various time levels. It highlights its computational efficiency and fast convergence using only a few series terms. Furthermore, numerical accuracy and stability of method are validated through error trend analysis and statistical correlation matrices for bright and dark soliton profiles. The results confirm the strong agreement between analytical and approximate solutions and support reliability and robustness of SHPM to solve nonlinear evolution equations relevant to optical fiber technology. The novelty of this work lies in the application of the SHPM to complex optical soliton models, providing an efficient and accurate alternative to conventional numerical methods with statistical significance of the fetched results.