A nonlinear model of diffusive particle acceleration at a planar shock

We study the process of nonlinear shock acceleration based on a nonlinear diffusion–advection equation. The nonlinearity is introduced via a dependence of the spatial diffusion coefficient on the distribution function of accelerating particles. This dependence reflects the interaction of energetic particles with self-generated waves. After thoroughly testing the grid-based numerical setup with a well-known analytical solution for linear shock acceleration at a specific shock transition, we consider different nonlinear scenarios, assess the influence of various parameters, and discuss the differences of the solutions to those of the linear case. We focus on the following observable features of the acceleration process, for which we quantify the differences in the linear and nonlinear cases: (1) the shape of the momentum spectra of the accelerated particles, (2) the time evolution of the solutions, and (3) the spatial number density profiles.