  The nonlinear dispersive wave equation inside the cylindrical elastic rod is
derived by applying the Navier-Bernoulli hypothesis and Love's relation in
\cite{5}. The elastic rod is assumed to be composed of the Murnaghan's
materials such as Lam$\acute{e}$'s coefficient, Poisson ratio and constitutive
constant which are compressible in nature. In this research paper we apply the
two integral architectures namely extended sine-Gordon method and modified
exponential function method to study the dispersive wave and solved for the
solitons and their classifications. The existence of the number of solutions
are proved with respect to the linear equation obtained by balancing principle.
The related two and three dimensional graphs are simulated and drawn to show
the complex structures.
