Wave-like avalanche propagation in the continuum field model of self-organized criticality

Travelling wave is identified as the mechanism of avalanche propagation in the continuum SOC (self-organized critical) system. Recovering the hidden causality based on a generalization of Fick's law, we lead the equivalent continuum equation which has spatiotemporal nonlocality. Taking into account of the hyperbolicity from the retarded response of the nonlocal kernel, it is possible to capture the propagating avalanche in constant speed. Verifying the computation, we analyze the evolution of instability as a nonlinear wave under the control of the threshold dynamics, which imposes the "metastability" as the survival condition of avalanche. Being consistent with the basic assumptions of SOC theory, the result shows unified features of the existing models. This provides a concept toward the SOC framework based on the physical principles of transport process.