Exploring the Jacobi elliptic and soliton solutions for the nonlinear (1+1)-dimensional Shynaray-IIA equation with conformable derivative

In this paper, we study the application of the Jacobi elliptic function expansion method for finding the new exact solutions to the nonlinear $(1+1)-$ dimensional Shynaray-IIA equation, including the conformable derivative. The Shynaray-IIA equation takes significant applications in fluid dynamics, plasma physics, and optical fibers, where understanding wave propagation is essential. Exact solutions are crucial because they provide deeper into the fundamental mechanisms, insights and physical phenomena described by the governing equations. The Jacobi elliptic function method offers solutions in the form of general trigonometric and hyperbolic functions for the applied equations. The derived solution could exhibit different wave structures such as rogue waves, bright and dark solitary wave structures. The physical importance of the derived solutions is fully highlighted in the areas of applications of the considered model. Moreover, some solutions are graphically plotted to showcase different wave structures using 3D, 2D and contour plots. The simplicity and usefulness of Jacobi elliptic techniques extend beyond the Shynaray-IIA equations to other challenging partial differential equations in science and technology.