Investigation of Novel Symmetry Solutions and Conservation Laws of a Generalized Double Dispersion Equation in Uniform and Inhomogeneous Murnaghan's Rod

Dispersion is a phenomenon in which a wave's phase velocity varies with its frequency. It can be relevant to all kinds of wave movements, including sound, seismic waves, and gravitational waves. The double dispersion equation is important due to its numerous physical applications, such as examining the nonlinear wave distribution in waveguides, investigating the interaction of waveguides with the surrounding medium, and assessing the probability of energy transfer through lateral waveguide coverings. In view of this, this article explores analytical examinations of a (1+1)-dimensional generalized double dispersion equation in inhomogeneous and uniform Murnaghan's rod. This is applicable in modeling wave propagation in an elastic solid material, which has significance in solid-state mechanics. Therefore, it is entrenched in solid-state physics. Lie group theory is invoked to identify point symmetries associated with the model, enabling the derivation of nonlinear ordinary differential equations through symmetry reduction. Furthermore, direct integration of the nonlinear ordinary differential equation is performed to obtain closed-form solutions to the underlying model. Consequently, an elliptic cosine function solution is attained. Additionally, using a specific transformation, the technique further ensures the attainment of a Weierstrass function solution. To secure more solutions to the studied equation, the well-known Kudryashov's method is utilized, affording us the opportunity to obtain an exponential function solution. Subsequently, we applied the (G′/G)-expansion technique, which consequently produces hyperbolic, rational, and trigonometric function solutions. Moreover, to view the wave dynamics of the achieved solutions, which provides us with the opportunity to capture the physical meanings of these solutions, various wave depictions are demonstrated in three-dimensional, two-dimensional, contour, and density plots. In conclusion, the study produces notable conserved quantities such as energy, mass, and momentum, which are secured using Ibragimov's theorem, as well as the multiplier approach.