This paper investigates an emergent equation known as the \(\mathfrak {q}\) -deformed equation or the \(\mathfrak {q}\) -deformed tanh-Gordon equation. We can better comprehend broken symmetries in physical systems by applying the \((\frac{\mathcal {Q}^{\prime }}{k \mathcal {Q}^{\prime }+\mathcal {Q}+r})\) -expansion technique to solve the q-deformed equation for definite parameter values. The solutions generated by this method provide significant insights into understanding the system’s dynamics and behavior. We employ the finite difference technique (FDM) to solve the \(\mathfrak {q}\) -deformed equation numerically and to verify the accuracy and dependability of our findings. Accurate and dependable results have been guaranteed by using this double technique. Additionally, tables and diagrams have been used to provide clarity and the ability to compare solutions from an analytical and numerical point of view. Through the use of these tools, readers are better equipped to understand how these tactics differ and concur. The \(\mathfrak {q}\) -deformation is an essential tool because it allows for a more accurate and nuanced representation of real-world occurrences by elucidating physical systems that exhibit non-traditional symmetry features like extensivity. Equation is very important in many subjects and areas because they make it easier to understand how complex physical systems are. By demonstrating the validity of the \(\mathfrak {q}\) -deformed equation in modeling systems with broken symmetries, this study has offered significant insights into the powers of the equation. We obtain inclusive solutions by the use of analytical and numerical techniques, and graphs have been used to guarantee their validity, precision and dependability. Investigating \(\mathfrak {q}\) -deformation advances modeling techniques and improves the representation of real-world processes with non-standard symmetry features.
