On the role of injection in kinetic approaches to nonlinear particle   acceleration at non relativistic shocks
  The dynamical reaction of the particles accelerated at a shock front by the first order Fermi process can be determined within kinetic models that account for both the hydrodynamics of the shocked fluid and the transport of the accelerated particles. These models predict the appearance of multiple solutions, all physically allowed. We discuss here the role of injection in selecting the {\it real} solution, in the framework of a simple phenomenological recipe, which is a variation of what is sometimes referred to as {\it thermal leakage}. In this context we show that multiple solutions basically disappear and when they are present they are limited to rather peculiar values of the parameters. 
Stefano Gabici (MPIK), Pasquale Blasi (INAF/Arcetri) and Giulia   Vannoni (MPIK)