Recent observations from satellites crossing active magnetic field lines have revealed solitary potential structures that move at speeds substantially greater than the ion thermal velocity. The structures appear as positive potential pulses rapidly drifting along the magnetic field. We interpret them as BGK electron holes supported by a population of trapped and passing electrons. Using Laplace transform techniques, we analyze the behavior of one phase-space electron hole. The resulting potential shapes and electron distribution functions are self-consistent and compatible with the field and particle data associated with the observed pulses. In particular, the spatial width increases with increasing amplitude. The stability of the analytic solution is tested by means of a two-dimensional particle-in-cell simulation code with open boundaries. We also use our code to briefly investigate the influence of the ions. The nonlinear structure appears to be remarkably resilient. ¿ 1999 American Geophysical Union