Influence of Truncated M-Fractional Derivative on Soliton Dynamics and Stability Analysis of Fifth-Order KdV Equation Using Improved Modified Extended Tanh Function Method

In this study, we explore the soliton solutions of the truncated M-fractional fifth-order Korteweg–de Vries (KdV) equation by applying the improved modified extended tanh function method (IMETM). Novel analytical solutions are obtained for the proposed system, such as brigh soliton, dark soliton, hyperbolic, exponential, Weierstrass, singular periodic, and Jacobi elliptic periodic solutions. To validate these results, we present detailed graphical representations of selected solutions, demonstrating both their mathematical structure and physical behavior. Furthermore, we conduct a comprehensive linear stability analysis to investigate the stability of these solutions. Our findings reveal that the fractional derivative significantly affects the amplitude, width, and velocity of the solitons, offering new insights into the control and manipulation of soliton dynamics in fractional systems. The novelty of this work lies in extending the IMETM approach to the truncated M-fractional fifth-order KdV equation for the first time, yielding a wide spectrum of exact analytical soliton solutions together with a rigorous stability analysis. This research contributes to the broader understanding of fractional differential equations and their applications in various scientific fields.