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A model is presented for a new type of fast solitary waves which is observed in downward current regions of the auroral zone. The three-dimensional, coherent structures are electrostatic, have a positive potential, and move along the magnetic field lines with speeds on the order of the electron drift. Their parallel potential profile is flattened and cannot fit to the Gaussian shape used in previous work. We develop a detailed BGK model which includes a flattened potential and an assumed cylindrical symmetry around a centric magnetic field line. The model envisions concentric shells of trapped electrons slowly drifting azimuthally while bouncing back and forth in the parallel direction. The electron dynamics is analysed in terms of three basic motions that occur on different time scales characterized by the cyclotron frequency <font face='Symbol'>W</font>_e , the bounce frequency <font face='Symbol'>w</font>_b , and the azimuthal drift frequency <font face='Symbol'>w_g</font>. The ordering <font face='Symbol'>W</font>_e >> <font face='Symbol'>w</font>_{b >>} <font face='Symbol'>w_g</font> is required. Self-consistent distribution functions are calculated in terms of approximate constants of motion. Constraints on the parameters characterizing the amplitude and shape of the stretched solitary wave are discussed.