Automodel solutions for superdiffusive transport by Lévy walks

Abstract

The method of approximate automodel solution for the Green’s function of time-dependent superdiffusive (nonlocal) transport equations (Kukushkin A B and Sdvizhenskii P A 2016 J. Phys. A: Math. Theor. 49 255002) is extended to the case of a finite velocity of carriers. This corresponds to an extension from Lévy flight based transport to transport of a type which belongs to the class of ‘Lévy walks + rests’, to allow for retardation effects in Lévy flights. This problem covers the cases of transport by resonant photons in astrophysical gases and plasmas, heat transport by electromagnetic waves in plasmas, migration of predators, and other applications. We treat a model case of 1D transport on a uniform background with a simple power-law step-length probability distribution function (PDF). A solution for the arbitrary superdiffusive PDF is suggested, and verification of the solution for a particular power-law PDF, which corresponds, e.g., to Lorentzian wings of atomic spectral line shapes for the emission of photons, is carried out using the computation of the exact solution.