We apply an Eulerian Vlasov code to study the amplification of an ultra-short seed pulse via stimulated Raman and Brillouin backscattering of energy from a long pump pulse, assumed at constant amplitude, in a plasma embedded in an external magnetic field. Detailed analysis of the spectra developed during the amplification process are presented, together with the evolution showing the pump depletion, accompanied by the counter-propagating seed-pulse amplification, compression and increased steepness of the waveform. In addition to the problem of the amplification of ultra-short seed pulses, there is an obvious academic interest in the study of problems of amplification of electromagnetic waves observed in many situations in laboratory plasmas and in the magnetosphere and other geophysical situations, such as in the environments of planets, where important variations in the presence and strength of magnetic fields are observed. The numerical code solves a one-dimensional relativistic Vlasov-Maxwell set of equations for a plasma in a magnetic field for both electrons and ions. We also apply the code to the problem of wakefield acceleration. The absence of noise in the Eulerian Vlasov code allows one to follow the evolution of the system with an accurate representation of the phase-space structures of the distribution functions.