Some Interesting Traveling Waves in a Transversely Isotropic Incompressible Hyperelastic Semi-infinite Rod

Abstract

In this paper, the bounded traveling wave solutions are examined for a system of partial differential equations with certain boundary conditions. Interestingly, this system can be used to describe the axially symmetric motion of a semi-infinite incompressible hyperelastic cylindrical rod. With the aid of traveling wave transformations, the partial differential equations can be reduced to a traveling wave equation. The implicit analytical expressions determining the traveling waves are derived. Significantly, the influences of material parameters on the qualitative properties are discussed in detail by using the phase portraits of the traveling wave system. Smooth solutions such as the periodic traveling wave solutions and the solitary wave solutions with the peak form are shown. In particular, some interesting singular traveling wave solutions including the solitary cusp wave solutions and the periodic cusp wave solutions with the peak form are obtained. Numerical examples for all these waves are given.