A soliton solution of the DD-Equation of the Murnaghan’s rod via the commutative hyper complex analysis

In this study, we apply the commutative hyper complex algebraic method to solve analytically the Doubly Dispersive Equation (DDE) in (1+1)-D, for the strain waves propagating in a cylindrical circular rod in complex polymer systems, within the framework of analytical mechanics. Relying on the stability of the solutions via phase portrait analysis, the resulting solitary waves obtained by applying the complex Jacobi elliptic function method, show the compression characters in the non-linearly elastic materials tackled in our article. The estimated Lagrangian density confirms the potential well criterion of the physical situation. Some comparisons are made with recent models conducted by several authors.