Nonlinear Thomson scattering of the ultrashort laser pulses

The process of the light scattering becomes nonlinear in the fields with relativistic intensity. Nonlinear scattering of ultrashort relativistic pulses by free electrons is considered. We use the wavelet basis in the form of derivatives from the Gaussian function for description of ultrashort pulses. The equation of motion, for a charged particle in the field of plane electromagnetic wave, has the exact solution including the expression for the energy. The approximation of instant spectrum allows to calculate the Thomson nonlinear scattering for laser pulses of ultrarelativistic intensity. Spectral distribution for all the pulse duration is the result of integration over the time. Exact solution in the case of laser pulse with form of the "Mexican hat" gives the velocity and acceleration in the parametrical form as functions of the proper time. The maximum of radiation for free electron in the fields with intensity 10(19) - 10(21) W/cm(2) is concentrated at the range of ultraviolet spectrum with photons energies 3-12 eV. The part of continuous spectrum reaches the area of large photon energies. One percent of scattered energy for the laser intensity 10(20) W/cm(2) is concentrated at the range (h) over bar omega > 2,7 . 10(2) eV, 10(21) W/cm(2) - (h) over bar omega > 7,9 . 10(3) eV, 10(22) W/cm(2) - (h) over bar omega > 2,45 . 10(5) eV. It allows one to use nonlinear scattering as a source of hard x-ray radiation.