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

TL;DR

This paper derives a nonlinear dispersive wave equation for a compressible elastic rod using classical hypotheses, and employs advanced methods to find and classify soliton solutions, supported by graphical simulations.

Contribution

It introduces a novel application of the extended sine-Gordon and modified exponential methods to soliton solutions in a compressible Murnaghan's material rod.

Findings

Existence of multiple soliton solutions proved.

Complex soliton structures visualized through 2D and 3D graphs.

Abstract

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.