We analyze the transverse stability for a configuration of multiple Gaussian bunches subject to the self-generated plasma wakefield. Through a semi-analytical approach we first study the equilibrium configuration for the modulated beam and then we investigate the evolution of the equilibrium configuration due to the emittance-driven expansion of the beam front that results in a rigid backward shift. The rear-directed shift brings the modulated beam out of the equilibrium, with the possibility for some of the bunch particles to be lost with a consequent deterioration of the driver. We look therefore for the proper position of the single bunches that maximize the stability without severely affecting the accelerating field behind the driver. We then compare the results with three-dimensional particle in cell simulations.

Plasma wakefield excited by a short TeV-scale proton beam is investigated in the highly nonlinear regime. Analysis of the “bubble” field illustrates that transverse expelling force of the wakefield can be compensated by the attractive force, which originates from the co-propagating electrons within the proton bunch, leading to a collimation effect that stabilizes the beam propagation. The protons located in the beam tail can be well-confined and accelerated forward for a long distance. Two-dimensional simulations show that after a 1-TeV proton bunch propagating through plasma for a distance, several percentages of the protons achieve a remarkable energy gain. This scheme presents a potential that proton beams from conventional accelerators may gain considerable additional energy through plasmas wakefields.