  The wake field excitation in an unmagnetized plasma by a multi-petawatt,
femtosecond, pancake-shaped laser pulse is described both analytically and
numerically in the regime with ultrarelativistic electron jitter velocities,
when the plasma electrons are almost expelled from the pulse region. This is
done, for the first time, in fluid theory. A novel mathematical model is
devised that does not break down for very intense pump strengths, in contrast
to the standard approach that uses the laser field envelope and the
ponderomotive guiding center averaging. This is accomplished by employing a
three-timescale description, with the intermediate scale associated with the
nonlinear phase of the electromagnetic wave and with the bending of its wave
front. The evolution of the pulse and of its electrostatic wake are studied by
the numerical solution in a two-dimensional geometry, with the spot diameter
\geq 100 microns. It reveals that the optimum initial pulse length needs to be
somewhat bigger than 1 micron (1-2 oscillations), as suggested by simple
analytical local estimates, because the nonlocal plasma response tends to
stretch very short pulses.
