  We propose the experiments on the collision of laser light and high intensity
electromagnetic pulses generated by relativistic flying mirrors, with electron
bunches produced by a conventional accelerator and with laser wake field
accelerated electrons for studying extreme field limits in the nonlinear
interaction of electromagnetic waves. The regimes of dominant radiation
reaction, which completely changes the electromagnetic wave-matter interaction,
will be revealed in the laser plasma experiments. This will result in a new
powerful source of ultra short high brightness gamma-ray pulses. A possibility
of the demonstration of the electron-positron pair creation in vacuum in a
multi-photon processes can be realized. This will allow modeling under
terrestrial laboratory conditions neutron star magnetospheres, cosmological
gamma ray bursts and the Leptonic Era of the Universe.
