The electric potential of tripolar spikes

We present an analytical formula for the waveform of the electric potential associated with a tripolar spike in a plasma. This formula is based on the construction and on the subsequent solution of a differential equation for the waveform. We work out this equation as a direct consequence of the morphological and functional properties of the observed waveform, without making any reference to the velocity distributions of the electrons and of the ions which sustain the spike. In the approximation of small potential amplitudes, we solve this equation by quadrature. In particular, in the second order approximation, the solution of this equation is given in terms of elementary functions. This analytical solution is able to reproduce the potential waveforms associated with electron holes, ion holes, monotonic and nonmonotonic double layers and tripolar spikes, in excellent agreement with observations.