Exploring optical soliton solutions of the time fractional q-deformed Sinh-Gordon equation using a semi-analytic method

The $\mathsf{q}$-deformed Sinh-Gordon equation extends the classical Sinh-Gordon equation by incorporating a deformation parameter $\mathsf{q}$. It provides a framework for studying nonlinear phenomena and soliton dynamics in the presence of quantum deformations, leading to intriguing mathematical structures and potential applications in diverse areas of physics. In this work, we imply the homotopy analysis method, to obtain approximate solutions for the proposed equation, the error estimated from the obtained solutions reflects the efficiency of the solving method. The solutions were presented in the form of 2D and 3D graphics. The graphics clarify the impact of a set of parameters on the solution, including the deformation parameter $\mathsf{q}$, as well as the effect of time and the fractional order derivative.