We investigate analytically and numerically the transport of cosmic rays following their escape from a shock or another localized acceleration site. Observed cosmic-ray distributions in the vicinity of heliospheric and astrophysical shocks imply that anomalous, superdiffusive transport plays a role in the evolution of the energetic particles. Several authors have quantitatively described the anomalous diffusion scalings, implied by the data, by solutions of a formal transport equation with fractional derivatives. Yet the physical basis of the fractional diffusion model remains uncertain. We explore an alternative model of the cosmic-ray transport: a nonlinear diffusion equation that follows from a self-consistent treatment of the resonantly interacting cosmic-ray particles and their self-generated turbulence. The nonlinear model naturally leads to superdiffusive scalings. In the presence of convection, the model yields a power-law dependence of the particle density on the distance upstream of the shock. Although the results do not refute the use of a fractional advection-diffusion equation, they indicate a viable alternative to explain the anomalous diffusion scalings of cosmic-ray particles.

We investigate analytically and numerically the effect of a time-dependent source in a nonlinear model of diffusive particle transport, based on the p-Laplacian equation. The equation has been used to explain the observed cosmic-ray distributions and it appears in fluid dynamics and other areas of applied mathematics. We derive self-similar solutions for a class of the particle source functions and develop approximate analytical solutions, based on an integral method. We also use the fundamental solution to obtain an asymptotic description of an evolving particle density profile, and we use numerical simulations to investigate the accuracy of the analytical approximations.

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.

We explore analytical techniques for modeling the nonlinear cosmic ray transport in various astrophysical environments which is of significant current research interest. While nonlinearity is most often described by coupled equations for the dynamics of the thermal plasma and the cosmic ray transport or for the transport of the plasma waves and the cosmic rays, we study the case of a single but nonlinear advection-diffusion equation. The latter can be approximately solved analytically or semi-analytically, with the advantage that these solutions are easy to use and, thus, can facilitate a quantitative comparison to data. In the present study, we extend our previous work in a twofold manner. First, instead of employing an integral method to the case of pure nonlinear diffusion, we apply an expansion technique to the advection-diffusion equation. We use the technique systematically to analyze the effect of nonlinear diffusion for the cases of constant and spatially varying advection combined with time-varying source functions. Second, we extend the study from the one-dimensional, Cartesian geometry to the radially symmetric case, which allows us to treat more accurately the nonlinear diffusion problems on larger scales away from the source.