Novel insights into the truncated M-fractional Shynaray-ⅡA equation: New wave solutions and its dynamic behavior

This article investigates the truncated M-fractional Shynaray-ⅡA equation, which has important applications in many fields, such as fiber optic communication, magnetodynamics of ferromagnetic materials, and fluid dynamics. First, the truncated M-fractional Shynaray-ⅡA equation is transformed into a nonlinear ordinary differential equation through a fractional-order derivative and the traveling wave transformation. Then, the generalized (G$ ^{'} $/G)-expansion method is adopted to get multiple groups of coefficient solutions based on the auxiliary equation. The conditions satisfied by the coefficients of the auxiliary equation are classified, and all novel wave solutions of the TMF-SⅡAE are obtained, including bright solitons, dark solitons, and breathers. Finally, in order to better understand the behavior of the solutions, a 3D graph, cartesian and polar graph, contour graph and density graph of the solutions are drawn using Python software. Specifically, we also conducted a comparative analysis between the newly obtained solution and the existing solution, and quantitatively discussed the parameters of fractional-order derivatives and analyzed their physical applications.