Non-linear cascade scattering of intense, tightly focused laser pulses by relativistic electrons is studied numerically in the classical approximation including the radiation damping for the quantum parameter hwx-ray/E<1 and an arbitrary radiation pa
rameter Kai. The electron energy loss, along with its side scattering by the ponderomotive force, makes the scattering in the vicinity of high laser field nearly impossible at high electron energies. The use of a second, co-propagating laser pulse as a booster is shown to solve this problem.
An external static magnetic field with its strength B~10T may result in the laser wake wave-breaking upon changing the electron motion in the vicinity of maximal density ramp of a wave period. This, as shown by numerical simulations, can change the r
esonance character of the electron self-injection in the laser wake-field; a total charge loaded in the acceleration phase of laser pulse wake can be controlled by a proper choice of the magnetic field strength.
