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.