Discrete Self-Similarity in Type-II Strong Explosions

TL;DR

This paper introduces new solutions for strong explosions in non-power law density profiles, revealing discrete self-similarity due to log-periodic perturbations, and verifies these solutions through numerical simulations.

Contribution

It presents the first analysis of discrete self-similarity in Type-II strong explosion solutions with log-periodic density perturbations.

Findings

Perturbations exhibit discrete self-similarity due to log-periodic density variations.

Numerical simulations confirm the analytical solutions.

Method can be generalized to arbitrary small, spherically symmetric perturbations.

Abstract

We present new solutions to the strong explosion problem in a non-power law density profile. The unperturbed self-similar solutions discovered by Waxman & Shvarts describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-\omega}$, where $\omega>3$ (Type-II solutions). The perturbations we consider are spherically symmetric and log-periodic with respect to the radius. While the unperturbed solutions are continuously self-similar, the log-periodicity of the density perturbations leads to a discrete self-similarity of the perturbations, i.e. the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. Finally we show that this method can be generalized to treat any small, spherically symmetric density perturbation by employing Fourier decomposition.