Simplification of Lie's Similarity Transformation Method of Solving Coupled Nonlinear Partial Differential Equations and Exact Dyon Solutions SU(2)Yang-Mills-Higg's Field Equations

For any \(\mathrm{n}\) number of coupled nonlinear partial differential equations for spherically symmetric field equations of the typer \({ }^2\left(\frac{\partial^2 \phi_j}{\partial r^2}-\frac{\partial^2 \phi_j}{\partial t^2}\right)=F \_i(\phi \mathrm{j})\), where \(\mathrm{j}=1,2, \ldots \mathrm{n}\), are the number of dependent variables and \(\mathrm{F} \mathrm{i}\left(\phi \_\mathrm{j}\right)\) are any functions of dependent variables \(\phi_j, \mathrm{j}=1,2, . . \mathrm{n}\). and free of independent variables \(\mathrm{r}\) and \(\mathrm{t}\) then a similarity variable is found as \(\mathrm{s}(\mathrm{r}, \mathrm{t})=\mathrm{r} /\left[\left(r^2-t^2\right)-\kappa t / \tau+\kappa^2 /\left(4 \tau^2\right)\right]\), where \(\kappa\) and \(\tau \neq 0\) are arbitrary integration constants. Using \(s(r, t)\) above coupled partial differential equations can be transformed into coupled ordinary differential equations. This result may reduce lengthy calculations for finding similarity transformations of coupled partial differential equations. Using this similarity variable two exact Dyon solutions of spherically symmetric Yang-Mills-Higg's field equations are found with 'circular functions.' For which known solutions are with hyperbolic functions.