FAST observations of Debye-length-scale potential structures are ubiquitous in the auroral return current region of the magnetosphere. We show that the 3D shape of these fast-moving coherent structures can be correctly reproduced with a nonlinear fluid model of a 3D electron-acoustic beam soliton. This model describes the early stage of the nonlinear beam-plasma instability which takes place on the high-potential side of an electrostatic potential ramp, where cold ionospheric electrons are accelerated up the magnetic field lines by a localized DC parallel electric field. At FAST altitudes (below 4000 km), our model predicts spheroidal potential structures, while at higher altitudes we expect the solitary waves to be elongated across the magnetic field. This altitude-dependent scaling is in agreement with observations of solitary waves performed with POLAR over a wide range of altitudes.