Investigation of complex hyperbolic and periodic wave structures to a new form of the q-deformed sinh-Gordon equation with fractional temporal evolution

This paper presents the fractional generalized q-deformed sinh-Gordon equation. The fractional effects of the temporal derivative of the proposed model are studied using a conformable derivative. The analytical solutions of the governing model depend on the specified parameters. The resulting equation is studied with two integration architectures: the sine-Gordon expansion method and the modified auxiliary equation method. These strategies extract hyperbolic, trigonometric, and rational form solutions. For appropriate parametric values and different values of fractional parameter α, the acquired findings are displayed via 3D graphics, 2D line plots, and contour plots. The graphical simulations of the constricted solutions depict the existence of bright soliton, dark soliton, and periodic waves. The considered model is useful in describing physical mechanisms that possess broken symmetry and incorporate effects such as amplification or dissipation.