In this paper, a nonlinear analytical solution for a cold plasma-bunched beam system based on the Hamiltonian formalism where a = ndno and fi+ = U& have been taken as parameters matching between zero and unity is given. The oscillation limiting energies, frequencies and transformer ratios have been carried out in general for both the one-dimensional and the case where a small transverse component of motion is included. The plasma wakefield accelerator has been treated as a special case.

Detailed electromagnetic analysis of the wake-field accelerator is presented. Particular attention is given to the two-dimensional dynamics model to understand how the wake-field affects the self-consistent radial dynamics of the beam. The corrent fields, in zeroth and first order, are obtained by using Fourier transform in the radial field. The self-focusing of the driving beam in a plasma is a result of the transverse wake. The ratio of the transverse wake-field to the longitudinal wake-field within the beam is calculated for small and large radii beams.PACS 52.40.Mj -Particle beam interactions in plasma (including intense charged-particle beams).