The Doubly Dispersive Equation (DDE) finds extensive utility across scientific and engineering domains. It stands as a significant nonlinear physical model elucidating nonlinear wave propagation within the elastic inhomogeneous Murnaghan’s rod (EIMR). With this in mind, we have focused on the integration of the DDE model and the advanced auxiliary equation (AAM) scheme. Through wave transformation, this model is effectively converted into an ordinary differential equation. In this paper, the goal of our work is to explore new wave solutions of the DDE model by using AAE scheme, which solutions are extremely helpful insights into the operation of the system. The impacts of the parameters are provided in this manuscript. We also discussed about the dynamical properties of the model, which is accomplished through bifurcation and stability investigations and also found the Hamiltonian function. This research makes a substantial contribution to the area by increasing our understanding of soliton solutions in the DDE, introducing novel investigation tools, and carrying out an in-depth investigation of the bifurcation and stability aspects of the system. As a direct result of this research, novel openings have been uncovered for further investigation and application in the various disciplines of science and engineering.