Solitons of nonlinear dispersive wave steered from Navier-Bernoulli hypothesis and Love's hypothesis in the cylindrical elastic rod with compressible Murnaghan's materials.

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.