  Presently available high-energy proton beams in circular accelerators carry
enough momentum to accelerate high-intensity electron and positron beams to the
TeV energy scale over several hundred meters of the plasma with a density of
about 1e15 1/cm^3. However, the plasma wavelength at this density is 100-1000
times shorter than the typical longitudinal size of the high-energy proton
beam. Therefore the self-modulation instability (SMI) of a long (~10 cm) proton
beam in the plasma should be used to create the train of micro-bunches which
would then drive the plasma wake resonantly. Changing the plasma density
profile offers a simple way to control the development of the SMI and the
acceleration of particles during this process. We present simulations of the
possible use of a plasma density gradient as a way to control the acceleration
of the electron beam during the development of the SMI of a 400 GeV proton beam
in a 10 m long plasma. This work is done in the context of the AWAKE project
--- the proof-of-principle experiment on proton driven plasma wakefield
acceleration at CERN.
