Investigating analytical and numerical techniques for the $$(2+1) {\mathfrak {q}}$$ -deformed equation

This paper presents a comprehensive study of a model called the \((2+1) {\mathfrak {q}}\)-deformed tanh-Gordon model. This model is particularly useful for studying physical systems with violated symmetries, as it provides insights into their behavior. To solve the \((2+1) {\mathfrak {q}}\)-deformed equation for specific parameter values, the \(({\mathfrak {H}}+\frac{{\mathcal {G}}^{\prime }}{ {\mathcal {G}}^{2}})\)-expansion approach is employed. This technique generates analytical solutions that reveal valuable information about the system’s dynamics and behavior. These solutions offer insights into the underlying mathematics and deepen the understanding of the system’s properties. To validate the accuracy of the analytical solutions, the finite difference technique is also used to find a numerical solution to the \({\mathfrak {q}}\)-deformed equation. This numerical approach ensures the correctness of the solutions and enhances the reliability of the results. Tables and graphics are presented in the publication to aid comprehension and comparison. These visuals improve the clarity and interpretability of the data, allowing readers to better understand the similarities and differences between the analytical and numerical solutions.