Nanjing Univ, Dept Astron, Nanjing 210093, Peoples R China
; Nanjing Univ, Astron & Astrophys Ctr E China, Nanjing 210093, Peoples R China
; Chinese Acad Sci, IHEP, Beijing 100039, Peoples R China
Based on the refined dynamical model proposed by us earlier for beamed gamma-ray burst ejecta, we carry out detailed numerical procedure to study those gamma-ray bursts with rapidly fading afterglows (i.e., similar to t(-2)). It is found that optical afterglows from GRB 970228, 980326, 980519, 990123, 990510 and 991208 can be satisfactorily fitted if the gamma-ray burst ejecta are highly collimated, with a universal initial half opening angle theta(0) similar to 0.1. The obvious light curve break observed in GRB 990123 is due to the relativistic-Newtonian transition of the beamed ejecta, and the rapidly fading optical afterglows come from synchrotron emissions during the mildly relativistic and non-relativistic phases. We strongly suggest that the rapid fading of afterglows currently observed in some gamma-ray bursts is evidence for beaming in these cases.