1D Longitudinal Beam Dynamics of Laser Plasma Wakefield Accelerator
  In this paper, we get the 1D approximate analytical solution of the plasma electrostatic wake driven by the laser, and get the oscillating frequency of the wake modified by the nonlinear high order terms, and find that the frequency depends on the oscillating amplitudes of the wake and the laser which is the general nonlinear phenomenon. Finally we analyze the longitudinal beam dynamics in this electrostatic wake, and find that the high order terms don't change the topology of the longitudinal phase space. 
Xiongwei Zhu
physics.acc-ph