Exploration of Soliton Solutions for the Kaup–Newell Model Using Two Integration Schemes in Mathematical Physics

Abstract

This research deals with the Kaup–Newell model, a class of nonlinear Schrödinger equations with important applications in plasma physics and nonlinear optics. Soliton solutions are essential for analyzing nonlinear wave behaviors in different physical systems, and the Kaup–Newell model is also significant in this context. The model's ability to represent subpicosecond pulses makes it a significant tool for the research of nonlinear optics and plasma physics. Overall, the Kaup–Newell model is an important research domain in these areas, with ongoing efforts focused on understanding its various solutions and potential applications. A new version of the generalized exponential rational function method and $\left(\frac{G^{\prime }}{G^2}\right)$-expansion function method are utilized to discover diverse soliton solutions. The generalized exponential rational function method facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to $\left(\frac{G^{\prime }}{G^2}\right)$-expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability of the proposed model is examined, with numerical simulations complementing the analytical results to provide a better understanding of the solutions' dynamic behavior. These results offer a foundation for future research, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations; to our knowledge, for this equation, these methods of investigation have not been explored before. The accuracy of each solution has been verified using the Maple software program.