A nonlinear energetic particle diffusion model with a variable source

Abstract

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.

A perturbative approach to a nonlinear advection-diffusion equation of particle transport

“…Since this topic has received considerable attention in recent years, analytical and semi-analytical results can serve as valuable guides for further more comprehensive modeling efforts. Consequently, we have expanded our previous work (Litvinenko et al, 2017;Litvinenko et al, 2019) in two ways.…”

“…On the other hand, we have complemented these analytical considerations with numerical simulations to investigate the influence of an additional advection of the background plasma on the particle distribution function. While this first study was limited to source functions being constant with respect to time, in the work of Litvinenko et al (2019), we have extended this analysis, for the case of pure diffusion, to time-variable sources.…”

“…We expand our previous work in a twofold manner. First, instead of employing an integral method, as in Litvinenko et al, (2019) to the case of pure nonlinear diffusion, we apply an expansion technique to the advection-diffusion equation to derive linear equations that are semi-analytically solvable with fundamental solutions. This allows an improved systematic analysis of the effect of nonlinear diffusion for the cases of constant as well as spatially varying advection, both with time-varying source functions.…”

“…The nonlinearity is most often described by coupled equations for the dynamics of the thermal plasma and the cosmic ray transport (e.g., Wiener et al, 2019) or for the transport of the plasma waves and the cosmic rays (e.g., Nava et al, 2016). Since these models require numerical solution procedures, we have recently investigated analytical and semi-analytical treatments on the basis of nonlinear diffusion equations (Litvinenko et al, 2017;Litvinenko et al, 2019) with the motivation to provide a treatment of the problem of nonlinear cosmic ray transport. These alternatives, which are based on a single advection-diffusion equation with a diffusion coefficient depending on the (gradient of the) particle distribution function, not only complement the development of the more detailed numerical models but also may guide as well as help to test the latter.…”

“…These alternatives, which are based on a single advection-diffusion equation with a diffusion coefficient depending on the (gradient of the) particle distribution function, not only complement the development of the more detailed numerical models but also may guide as well as help to test the latter. While in the work of Litvinenko et al (2017) we have concentrated on the implications of the nonlinearity for the so-called anomalous transport (for a review of applications see dos Santos, 2019) in one-dimensional, Cartesian advective-diffusive systems, in the work of Litvinenko et al (2019), we have extended the modeling to time-varying source functions in the absence of advection. We extend these analyses here to systems with radial symmetry and with time-dependent sources as well as nonvanishing advection, which are often of interest in astrophysics, for instance, for the particle transport in the solar wind.…”