On the exact soliton solutions and different wave structures to the double dispersive equation

In this manuscript, we are interested to investigate the dynamical behavior of doubly dispersive equation which governs the propagation of nonlinear waves in the elastic Murnaghan’s rod. A variety of solitary wave solutions with unknown parameters is extracted in different shapes such as bright, dark, kink-type, bell-shape, combine and complex soliton, hyperbolic, exponential, and trigonometric function solutions by the employing recently developed methods, namely, new extended direct algebraic method (NEDAM) and generalized Kudryashov method (GKM). Furthermore, we apply the logarithmic transformation, and the ansatz functions method along with symbolic computation. The three waves, double exponential, and homoclinic breather techniques are also utilized to observe the soliton’s interaction phenomenon. A conflict of our results with the considerably-known results are done and it the study states that the solutions reached here are new. In addition, 3D, 2D, and their crossponding contour profiles of earned results are sketched in order to observe their dynamics with the choices of involved parameters. On the bases of achieved results, we may claim that the proposed computational method are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations. The results obtained for the governing model can be applied to other nonlinear equations. The findings are exceptional and unique in comparison to previous findings in the literature. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the solids. We feel that this work is timely and will be of interest to a wide spectrum of experts working on engineering models.