Selfacceleration of electrons in one-dimensional bunches, moving in cold plasma

Nonlinear dynamics of the one-dimensional ultrarelativistic bunch of electrons, moving in cold plasma, is considered in multiple scales perturbative approach. A square root of the inverse Lorentz factor of the bunch electrons is taken as a small parameter. Bunch electrons momenta is changed in the first approximation. In the underdense plasma and for the model example of the combined bunch the selfacceleration of the bunch electrons can be remarkable.