Kinetic approaches to particle acceleration at cosmic ray modified shocks

Kinetic approaches provide an effective description of the process of particle acceleration at shock fronts and allow us to take into account the dynamical reaction of the accelerated particles as well as the amplification of the turbulent magnetic field as due to streaming instability. The latter does in turn affect the maximum achievable momentum and thereby the acceleration process itself, in a chain of causality which is typical of non-linear systems. Different kinetic approaches are characterized by different levels and types of approximations that also imply different computational times. Here we present the results of two such approaches: one which is mathematically rigorous but rather demanding from the point of view of computational time, and the other which is computationally very fast but based on an ansatz that, while physically justified, is not rigorous. The identification of possible differences can be crucial in assessing the possibility of implementation of one such calculation in hydrodynamical codes for supernova explosions. Special emphasis is given to a discussion of the appearance of multiple solutions in both approaches.