Analytical soliton-like solutions of the electromagnetic and scalar fields equation in minimal coupling

Taking into account the proper gravitational field of the elementary particles, we have firstly determined the exact, static, spherical, symmetric solutions of the nonlinear electromagnetic and scalar fields equation in minimal coupling, considering all forms of the function S ( k , ξ ) . It is showed that all metrics functions and solutions of the interacting field equation are regular with the localized energy density. The total energy of interacting fields E f is finite and the total charge of the particles Q is finite. Finally, the stability analysis of electric scalar potential and metric functions is studied. As results, it is found that all the studied configurations are stable on the range [ 0 , ξ c ] .