Invariance properties of the microstrain wave equation arising in microstructured solids

The Lie group method stands as a significant, powerful, and straightforward mathematical tool for discovering precise invariant solutions and traveling wave solutions in prevalent nonlinear evolution equations across engineering, applied mathematics, and physics. The interplay of dispersive effects arising from material microstructure, combined with nonlinearities, results in the formation of solitary waves. In this article, we apply the Lie group method to derive invariant and comprehensive traveling wave solutions for the strain wave equation in microstructured solids. This yields a diverse array of exact solutions, including traveling wave solutions, solitons, shock waves, periodic, singular, and rational solutions, all of which bear potential significance in various engineering applications.