In this research, we apply some new mathematical methods to the study of solving couple-breaking soliton equations in two dimensions. Soliton solutions for equations with free parameters like the wave number, phase component, nonlinear coefficient and dispersion coefficient can be obtained analytically by adding trigonometric, rational and hyperbolic functions. We will also look into how two-dimensional diagrams are affected by the wave phenomena, illustrating the answers with a mix of two- and three-dimensional graphs. The proposed system will be transformed into a numerical system by using the finite difference method to simulate Black-Scholes equations numerically. Furthermore, we will evaluate the stability and accuracy of the numerical findings by making analytical and graphical comparisons with precise solutions and we will talk about the error analysis of the numerical scheme. All forms of nonlinear evolutionary systems can benefit from the methods utilized in this work because they are sufficient and acceptable.