Soliton-like solutions for a nonlinear doubly dispersive equation in an elastic Murnaghan's rod <i>via</i> Hirota's bilinear method

Abstract The purpose of this work is to study the physical phenomena of the doubly dispersive model that controls chaotic wave movement in the elastic Murnaghan’s rod. The method of Hirota bilinear transformation is employed to derive various forms of solitary wave solutions, such as multiple waves, periodic lump waves, periodic cross-kink waves, homoclinic breather waves, dark soliton, and mixed waves. In order to see their physical behavior, we use the Mathematica software with selected values of the model parameters to depict their graphical behavior.