The present work is attentive to studying the qualitative analysis for a nonlinear strain wave equation describing the finite deformation elastic rod taking into account transverse inertia, and shearing strain. The strain wave equation is rewritten as a dynamic system by applying a particular transformation. The bifurcation of the solutions is examined, and the phase portrait is depicted. Based on the bifurcation constraints, the integration of the first integral of the dynamic system along specified intervals leads to real wave solutions. We prove the strain wave equation has periodic, solitary wave solutions and does not possess kink (or anti-kink) solutions. In addition, the set of discovered solutions contains Jacobi-elliptic, trigonometric, and hyperbolic functions. This model contains many kinds of solutions, which are always symmetric or anti-symmetric in space. We study how the change in the physical parameters impacts the solutions that are found. Numerically, the behavior of the strain wave for the elastic rod is examined when particular periodic forces act on it, and moreover, we clarify the existence of quasi-periodic motion. To clarify these solutions, we present a 3D representation of them and the corresponding phase orbit.

Our study focuses on the governing equation of a finitely deformed flexible rod with strain waves. By utilizing the well-known Ablowita–Ramani–Segur (ARS) algorithm, we prove that the equation is non-integrable in the Painlevé sense. Based on the bifurcation theory for planar dynamical systems, we modify an auxiliary equation method to obtain a new systematic and effective method that can be used for a wide class of non-linear evolution equations. This method is summed up in an algorithm that explains and clarifies the ease of its applicability. The proposed method is successfully applied to construct wave solutions. The developed solutions are grouped as periodic, solitary, super periodic, kink, and unbounded solutions. A graphic representation of these solutions is presented using a $3D$ representation and a $2D$ representation, as well as a $2D$ contour plot.