Revisiting the Emission from Relativistic Blast Waves in a Density-Jump   Medium
  Re-brightening bumps are frequently observed in gamma-ray burst (GRB) afterglows. Many scenarios have been proposed to interpret the origin of these bumps, of which a blast wave encountering a density-jump in the circumburst environment has been questioned by recent works. We develop a set of differential equations to calculate the relativistic outflow encountering the density-jump by extending the work of Huang et al. (1999). This approach is a semi-analytic method and is very convenient. Our results show that late high-amplitude bumps can not be produced under common conditions, only short plateau may emerge even when the encounter occurs at early time ($< 10^4$ s). In general, our results disfavor the density-jump origin for those observed bumps, which is consistent with the conclusion drawn from full hydrodynamics studies. The bumps thus should be due to other scenarios. 
J. J. Geng, X. F. Wu, Liang Li, Y. F. Huang and Z. G. Dai
astro-ph.HE