In this study, the (3+1) dimensional generalized Yu-Toda-Sasa-Fukuyama equation (YTSF) and its relevance to physical and engineering systems are investigated. We apply the newly developed \(\phi ^6\) -model expansion method to reduce the YTSF equation to a set of ordinary differential equations, allowing for the construction of exact analytical solutions. The resulting soliton solutions, expressed in terms of Jacobi elliptic functions, include parabolic dark solitons, shock wave solutions, bell-shaped solitons, W-shaped solitons, U-shaped solitons, and smooth periodic solitons. These solutions are derived using Maple and are graphically illustrated through 2D, 3D, and contour plots with Mathematica. Solitons, known for their ability to retain shape and velocity over long distances, are ideal for data transmission in electric communication systems. To evaluate the robustness of the model, sensitivity analysis is performed, demonstrating how small changes in initial conditions can significantly influence system dynamics. This study is novel in applying the \(\phi ^6\) -model expansion method to the (3+1)-dimensional generalized YTSF equation for the first time, offering a diverse set of new soliton solutions with potential applications in the design and optimization of modern communication systems. It is very helpful for researchers and engineers to create electric communication systems and make better judgments.
