Non linear particle acceleration at non-relativistic shock waves in the   presence of self-generated turbulence
  Particle acceleration at astrophysical shocks may be very efficient if magnetic scattering is self-generated by the same particles. This nonlinear process adds to the nonlinear modification of the shock due to the dynamical reaction of the accelerated particles on the shock. Building on a previous general solution of the problem of particle acceleration with arbitrary diffusion coefficients (Amato & Blasi, 2005), we present here the first semi-analytical calculation of particle acceleration with both effects taken into account at the same time: charged particles are accelerated in the background of Alfven waves that they generate due to the streaming instability, and modify the dynamics of the plasma in the shock vicinity. 
Elena Amato (1), Pasquale Blasi (1) ((1) INAF - Osservatorio   Astrofisico di Arcetri)