Propagation of solitary wave in micro-crystalline materials

In this article, the propagation of waves in micro-crystalline materials is governed by the structure of the strain wave equation and takes into consideration various dimensions of micro-crystalline. Micro-crystalline materials that are deserving of special attention in material physics. The strain wave equation represents the dynamic behavior associated with multiple phenomena of a physical nature. The new extended direct algebraic methodology is applied to acquire the different types of exact solitonic solutions. This technique stands out as one of the most effective approaches for producing a diverse set of exact solutions to nonlinear partial differential equations. By applying a new extended direct algebraic approach, we get solutions in the form of smooth periodic, anti -dark, anti-bell-shape, periodic bright, Combined bright -dark soliton, mixed-periodic solution, anti -kink formations, Stumpons, mixed periodic solitons, and decaying cusped solitons. Furthermore, two-dimensional, three-dimensional, and contour plots are created for different solutions, helping us make sense of their physical significance more clearly. The importance of the obtained results lies in their ability to represent diverse and complex phenomena in mathematical and physical systems.