Wave Propagation and Soliton Behaviors for the Strain Equation by Using the Sub-ODE Method and Expansion Technique

Within this study, we use the exact techniques to compute the new distinct soliton solutions of the strain wave equation. The governing equation is very important because it is use to interpret wave propagation in microstructured solids. By the use of Sub-ODE technique, the different categories of solitons including periodic solitons, rational solitons, dark solitons, bright solitons, travelling waves, trigonometric, and Weierstrass elliptic function are obtained. The main purpose behind the sub-ODE method is to find the soliton solutions of a nonlinear model by using simple and solvable ODEs called sub-ODEs. Additionally, the various exact solutions in the form of rational, exponential, hyperbolic, and trigonometric functions are derived using the Exp \((-\phi (\eta ))\) $(-\varphi (\eta ))$ -expansion technique. Variants wave results are found from the exact propagating wave solution by changing the parameters to varied values. To better comprehend the physical phenomena of the micro-structured solids wave model, several solutions have been illustrated graphically. The findings demonstrate how the system parameters, which can be employed as system controllers, affect the wave solutions dynamics. According to these findings, the sub-ODE and Exp \((-\phi (\eta ))\) $(-\varphi (\eta ))$ -expansion methods are simpler, more efficient, and more powerful mathematical tools for determining the exact solitary solutions to nonlinear partial differential equations that occur in the fields of engineering, mathematics, physics, fluid mechanics, materials science, fiber optics, and many other natural sciences. Using mathematical software, the obtained results have been validated by inserting them into the governing equation. Therefore, our methods based on the foundation of representative computations provide a rigorous and efficient mathematical implementation for solving complex nonlinear wave problems.