  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 3D PIC simulations.
