Superdiffusive Transport Based on Levy Walks in a Homogeneous Medium: General and Approximate Self-Similar Solutions

The general and approximate self-similar solutions for the Green's function have been obtained for a wide class of integrodifferential equations for the two- and three-dimensional (in spatial coordinates) nonstationary superdiffusive transport of a perturbation of a homogeneous medium for a finite fixed velocity of carriers. This problem concerns the transport of resonance radiation in astrophysical gases and plasma, migration of animals, and the transfer of energy of electromagnetic waves in the plasma. The case of a model free path distribution function decreasing by a power law with increasing distance has been considered. Numerical calculations have been performed for two particular types of free path distribution functions, including the case with a Lorentzian shape of wings of the spectral line profile for the emission of photons by atoms or ions. The method developed in [A.B. Kukushkin and P.A. Sdvizhenskii, J. Phys. A: Math. Theor.49, 255002 (2016)] for an infinite velocity of carriers has been used to construct the approximate self-similar solution. The inclusion of a finite velocity corresponds to the generalization of transport based on Levy flights to transport based on "Levy walks with stops." The self-similar solution for arbitrary one-dimensional superdiffusive transport obtained in [A.B. Kukushkin and A.A. Kulichenko, Phys. Scripta94, 115009 (2019)] has been applied to the case of the two- and three-dimensional transports. The accuracy of the self-similar solution has been tested by comparing it to the numerically calculated general solution.