Solitons in an inhomogeneous Murnaghan’s rod

In this paper, we construct a family of wave solutions to the doubly dispersive equation, such as topological, non-topological, singular, compound topological-non-topological bell-type and compound singular, soliton-like, singular periodic wave and exponential function solutions. These analytical solutions are obtained by using the extended sinh-Gordon equation expansion method and the modified exp (- φ(ζ)) -expansion function method. The doubly dispersive equation is an important nonlinear physical model describing the nonlinear wave propagation in the elastic inhomogeneous Murnaghan’s rod. Under a suitable choice of parameters, the 2D, 3D and contour graphics to the reported results are also plotted.