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.
Highlights
F (x, t) is the energetic particle density, x is the distance along the mean magnetic field, t is time, and S = S(x, t) is a particle source
Concrete physical mechanisms were shown to yield Equation (1) with ν = 1/2 and ν = 2/3 (Ptuskin et al, 2008) under the simplifying assumption that the turbulent wave generation by the particles is balanced by wave dissipation: the case ν = 1/2 corresponds to wave energy transfer to the thermal ions that interact with moving magnetic mirrors formed by the waves (Zirakashvili, 2000), and the case ν = 2/3 corresponds to energy dissipation by a Kolmogorov-type energy cascade (Ptuskin and Zirakashvili, 2003)
We explored the roles of advection and a constant particle source and used the results to interpret the observed distributions of energetic ions at the termination shock of the solar wind (Litvinenko et al, 2017)
Summary
To obtain an approximate analytical solution of the nonlinear Equation (1), we use an integral method. Numerical solutions of Equation (1) suggest that the key effect of a time-dependent particle source in the nonlinear diffusion model is the localization of the particle distribution on a spatial scale w(t) that is controlled by the source strength S(t). Equations (5) and (8) show that the approximate solution for the particle density f (x, t) turns out to be a separable function: f (x, t) ≈
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