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The AWAKE experiment at CERN will be the first proof-of-principle demonstration of the proton-driven plasma wakefield acceleration using the 400 GeV proton beam extracted from the SPS accelerator. The plasma wakefield will be driven by a sequence of sub-millimeter long micro-bunches produced as a result of the self-modulation instability (SMI) of the 12 cm long SPS proton bunch in the 10 m long rubidium plasma with a density corresponding to the plasma wavelength of around 1 mm. A 16 MeV electron beam will be injected into the developing SMI and used to probe the plasma wakefields. The proton beam self-modulation in a wide range of plasma densities and gradients have been studied in detail via numerical simulations. A new configuration of the AWAKE experiment with a small plasma density step is proposed.
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