  For the D-branes on the SL(2,R) WZW model we present a particular choice of
outer automorphism for the gluing condition of currents that leads to a special
AdS_2 brane configuration. This configuration is shown to be a static solution
in the cylindrical coordinates, and a nonstatic solution in the Poincar\'e
coordinates to the nonlinear equation of motion for the Dirac-Born-Infeld
action of a D-string. The generalization of it gives a family of nonstatic
AdS_2 brane solutions. They are deomonstrated to transform to each other under
the isometry group of AdS_3 spacetime.
