Analytical and dynamical insights into the truncated M-fractional Akbota equation

This paper investigates the truncated M-fractional Akbota equation via the application of Jacobi elliptic function expansion approach. Akbota is an integrable Heisenberg ferromagnetic couple system of equations that is very useful to visualize and study for surface geometry, curve analysis in optics and magnets. The primary objective of this study is to construct new exact solutions and provide comprehensive visualizations of the Akbota equation based on the M-fractional derivative under investigation. By employing the proposed approach, a suitable traveling wave transformation is applied to reduce the governing nonlinear partial differential equation into an ordinary differential equation of integral order. Using a finite series expansion in terms of Jacobi elliptic functions, several new classes of exact solutions are derived. The obtained results include Jacobi elliptic, hyperbolic, and trigonometric function solutions, where the latter two are recovered as the modulus m approaches 1 and 0, respectively. The findings hold significant potential in various physical and engineering applications, such as telecommunications, nonlinear optics, and optical fiber systems. Furthermore, graphical illustrations of the solutions obtained via the Jacobi Elliptic Function Expansion (JEFE) method are presented to reveal the physical behavior of the model. Finally, a dynamical analysis within the framework of chaos theory is carried out, demonstrating how the system’s dynamics evolve under appropriate parameter choices. Overall, this study deepens the understanding of the Akbota equation and highlights the JEFE method as an effective and reliable tool for analyzing nonlinear dynamical systems.