In a test-particle (linear) acceleration theory, the shock compression controls the particle spectrum. We present a nonlinear self-similar solution in which the accelerated particles change the flow structure near the shock so strongly that the total shock compression r may become very high. Despite this, the energy spectrum remains close to E -3/2 independent of r . This result is valid for the particle diffusivity κ( p ) growing with momentum faster than p 1/2 , which is believed to be the case.
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Mikhail Malkov (1999) studied this question.