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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.

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations. [ABSTRACT FROM AUTHOR]