  A new regime of proton-driven plasma wakefield acceleration is discovered, in
which the plasma nonlinearity increases the phase velocity of the excited wave
compared to that of the protons. If the beam charge is much larger than
minimally necessary to excite a nonlinear wave, there is sufficient freedom in
choosing the longitudinal plasma density profile to make the wave speed close
to the speed of light. This allows electrons or positrons to be accelerated to
about 200 GeV with a 400 GeV proton driver.
