Some nonlinear partial differential equations and methods of exact solution

In this study, a fourth-order nonlinear partial differential equation modeling the propagation of light pulses in optical fibers is addressed. The equation exhibits a highly complex structure due to its inclusion of nonlinear terms and higher-order dispersion effects. It provides a significant mathematical framework for understanding data transmission, soliton formation, and the dynamic behavior of solitons in optical fiber systems. Consequently, obtaining exact solutions to the equation holds great importance both in theoretical and applied sciences. To derive exact solutions to the equation, three different methods widely recognized in the literature for their systematic and effective approach were employed: the Sardar sub-equation method, the Riccati-Bernoulli method, and the $\left(\dfrac{G'}{kG'+G+r}\right)$-expansion method. These techniques are powerful tools frequently used for solving nonlinear partial differential equations. Within the scope of this study, novel and original exact analytical solutions were obtained using these methods. The solutions not only provide valuable mathematical insights but also demonstrate applicability in various scientific fields, such as modeling soliton behavior in optical fibers, understanding the propagation of electromagnetic waves in plasma physics, and advancing fiber optic technologies. The physical significance of the analytical solutions was visualized through graphs of specific solutions. These visualizations play a critical role in explaining the spatiotemporal evolution of soliton behaviors. By emphasizing the practical accuracy and scientific importance of the solutions, these graphs contribute to a deeper understanding of nonlinear phenomena. In conclusion, this study offers exact solutions to a challenging equation and makes a substantial contribution to the literature on nonlinear partial differential equations and their diverse applications in scientific and engineering fields.