Hydromagnetic shock structure in the presence of cosmic rays

The time asymptotic structure of a shock significantly modified by the back-reaction from the diffusive acceleration of cosmic rays is investigated. Making a physically plausible assumption about the diffusion, it is shown that for given upstream conditions and shock speed only a finite odd number of shock structures are possible; an explicit method of determining these is given (in many cases the solution is unique). The results of this nonlinear study are contrasted with those of the linear test-particle theory and shown to confirm the possibility of efficient particle acceleration in shocks.