A non-linear equation for describing the dynamics of slowly varying envelope of ultra-intense lasers in electron-positron-ion plasmas is derived, taking into account the vacuum polarization effect. Then the modulation instability of ultra-intense laser pulses in the plasmas is investigated. Theoretical results show that the growth rate of modulation instability is originated from the relativistic ponderomotive force of lasers and the vacuum polarization effect. In the regime of different physical parameters, there is an interplay of the two effects. The dependence of the growth rate on the intensity of lasers, the temperature of electrons (positrons), the component of plasmas and the vacuum polarization effect is numerically studied.
Nonlinear theory of intense laser-plasma interactions modified by vacuum polarization effects