This research aims to investigate a generalized fifth-order nonlinear partial differential equation for the Sawada-Kotera (SK), Lax, and Caudrey-Dodd-Gibbon (CDG) equations to study the nonlinear wave phenomena in shallow water, ion-acoustic waves in plasma physics, and other nonlinear sciences. The Painlevé analysis is used to determine the integrability of the equation, and the simplified Hirota technique is applied to construct multiple soliton solutions with an investigation of the dispersion relation and phase shift of the equation. We utilize a linear combination approach to construct a system of equations to obtain a general logarithmic transformation for the dependent variable. We generate one-soliton, two-soliton, and three-soliton wave solutions using the simplified Hirota method and showcase the dynamics of these solutions graphically through interaction between one, two, and three solitons. We investigate the impact of the system’s parameters on the solitons and periodic waves. The SK, Lax, and CDG equations have a wide range of applications in nonlinear dynamics, plasma physics, oceanography, soliton theory, fluid dynamics, and other sciences.