Exploring unconventional optical soliton solutions for a novel $\mathfrak{q}$-deformed mathematical model

This paper presents a significant contribution in the form of a new general equation, namely the $\mathfrak{q}$-deformed equation or the $\mathfrak{q}$-deformed tanh-Gordon equation. The introduction of this novel equation opens up new possibilities for modeling physical systems that exhibit violated symmetries. By employing the $(G^{\prime }/G)$ expansion method, we have successfully derived solitary wave solutions for the newly defined $\mathfrak{q}$-deformed equation under specific parameter regimes. These solutions provide valuable insights into the behavior of the system and its dynamics. To further validate the obtained analytical results, the numerical solution of the $\mathfrak{q}$-deformed equation has been constructed by using the finite difference method. This numerical approach ensures the accuracy and reliability of the findings. To facilitate a comprehensive understanding of the results, we have included two- and three-dimensional tables and figures, which provide visual representations and comparisons between the analytical and numerical solutions. These graphical illustrations enhance the clarity and interpretation of the obtained data. The significance of the $\mathfrak{q}$-deformation lies in its ability to model physical systems that exhibit deviations from standard symmetry properties, such as extensivity. This type of modeling is increasingly relevant in various fields, as it allows for a more accurate representation of real-world phenomena.