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.