Retracted: Exploring new optical solutions for nonlinear Hamiltonian amplitude equation via two integration schemes

Abstract

This research explores the Jacobi elliptic expansion function method and a modified version of the Sardar sub-equation method to discover new exact solutions for the nonlinear Hamiltonian amplitude equation. {\color{blue}By applying these techniques, the study seeks to uncover previously unknown solutions for this equation, contributing to the understanding of its behavior and opening up new possibilities for its applications.} The solutions obtained using these methods are represented by hyperbolic, trigonometric, and exponential functions, and they include optical dark-bright, periodic, singular, and bright solutions. The dynamic behaviors of these solutions are demonstrated by selecting appropriate values for physical parameters. By assigning values to these parameters, the study aims to showcase how the solutions of the nonlinear Hamiltonian amplitude equation behave under different conditions. This analysis provides insights into the system's response and enables a deeper comprehension of its complex dynamics in various scenarios, contributing to the overall understanding of the equation's behavior and potential real-world implications. Overall, these methods are effective in analyzing and obtaining analytic solutions for nonlinear partial differential equations.