Analytical Study on the Generalized q-Deformed Sinh–Gordon (Eleuch) Equation

The mathematical definition and analysis of many physical events depend heavily on nonlinear models. These models are recognized as indispensable tools for understanding the intricacies of complex systems and predicting their future behaviors. These equations are mathematical models that hold significant importance in theoretical physics, mechanics, and quantum field theory. Solving these equations is crucial for understanding complex structures and discovering new phenomena. In this study, we investigate the exact solutions of a new form of the generalized q-deformed Sinh–Gordon (Eleuch) equation. To obtain solutions not found in the current literature for this equation, we apply a new auxiliary equation method that generates more solutions compared to other methods. Additionally, we demonstrate 3D, 2D, contour, and density plots to showcase different soliton propagation patterns. Through the solutions of this equation, we provide more options in the literature for simulating physical systems with broken symmetry.