Solitary Wave Solution of the Third Order Gilson-Pickering Equation Via the Abel’s Equation with Variable Coefficients

A novel analytical method, termed the Variable Coefficient Second Degree Generalized Abel Equation Method, is presented in this paper. It has been specifically developed to address the complexities of the Gilson Pickering equation—an important nonlinear partial differential equation encountered in various physical applications. This equation is recognized as a general case of several significant nonlinear partial differential equations, including the Fornberg Whitham, Rosenau Hyman, and Fuchssteiner-Fokas-Camassa-Holm equations. Unlike conventional approaches, which are typically based on constant coefficient ordinary differential equations (ODEs) or auxiliary equations, a unique framework based on variable coefficient ODEs integrated within a subequation structure is introduced by this method.. By applying the method to the Gilson Pickering equation, new exact analytical solutions are derived. The correctness of the technique is validated, and its superior efficiency and robustness are highlighted through these results. The intricate dynamics of the equation are effectively captured, demonstrating the method’s suitability for modeling phenomena in fluid dynamics, nonlinear optics, and wave propagation. Furthermore, the potential application of the method to a broader class of nonlinear partial differential equations across mathematical physics is implied by its generalizability. The importance of advancing analytical tools to better understand and resolve complex physical systems is emphasized by the findings, thereby marking a significant contribution to the analytical study of nonlinear differential equations. Ultimately, an expansion of the repertoire of solution strategies available to researchers in applied mathematics and theoretical physics is achieved through this work.