a region without sufficient scattering to return them they remove energy (and total pressure) from the shock. This likely eventuality further enhances the total compression through the shock, much like what happens in a radiative shock. On the other hand, some models explaining how particles manage to be “injected” into the CR population by escaping upstream through the shock for the first time would predict a reduced injection efficiency once the subshock is weakened. That, in turn, would reduce the expected $P_{cr}$, so provide feedback of the opposite sign. Malkov, (2000) have shown a bifurcation in the steady state analytic solutions when these various features are included and argue that a critical, self-organization may develop that determines the weights of the various nonlinear effects. Figure 2, from Kang, 2002, illustrates some these features in the evolution of a simulated CR shock.

There are still other nonlinear features of the DSA process, most of which are less well understood than what I just described. I refer interested readers to some of the recent reviews for further information (e.g., Malkov & Drury 2001; Jones 2001).

Especially in some phenomena considered for acceleration of UHECRs the flows into the shocks may be relativistic 