Search for approximate methods for description of nonlinear vacuum $e^-e^+$ pairs creation processes in electromagnetic fields

The simplest kinetic equation for description of the electron-positron plasma vacuum creation in a strong linearly polarized electric («laser») field was reduced to the nonlinear ordinary differential equation of the second order. The corresponding truncated equation without the dissipative contributions was obtained also. In area of the tunnel mechanism action the non-local under an external field solutions for the residual electron-positron plasma was first obtained. In general case, the upper estimations for the kinetic equation solutions was found on the Riccati equation basis that is a result of application of the stereographic projection method to the basic kinetic equation.