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Abstract

Stationary solutions in the non-linear theory of diffusive shock acceleration play a fundamental role in the theory of cosmic-ray acceleration. Their existence usually requires that a fraction of accelerated particles be allowed to escape from the system. This condition is incorporated into analyses of the kinetic equations by introducing an upper cut-off to the energy of such particles. However, the form of the distribution function close to the cut-off - a diagnostic that is accessible to observation - cannot be described using this method. We have developed an alternative approach in which particles escape over a boundary placed at a finite distance upstream of the shock front, and present results on the role of multiple solutions, and effects on the proposed self-organised criticality of the system.