Surface charge carrier motion on insulating surfaces: One dimensional motion

We consider a model for the motion of charge carriers on the surface of an insulator. The insulator surface is either infinite, semi–infinite against a conducting half space or a strip between two conducting half spaces. The charge flux on the surface is assumed equal to the charge density times the electric field component in the surface, with time a constant. When the charge carrier motion in the plane is assumed constant in one direction, we can write the problem as an inviscid Burgers equation for a complex function. The imaginary part of this function is minus the carrier density while the real part, the Hilbert transform of the carrier density, is minus the electric field on the surface. Using the method of characteristics, we find an exact implicit solution for the problem and illustrate it with several examples. One set of examples, on the real line, or half of it, show how charge moves and how the surface may discharge into a conducting wall. They also show that the system can sustain shock wave solutions which are different from those in a real Burgers equation and other singular behaviour. Exact solutions on a finite strip between two conducting walls also show how that system can discharge completely, and also demonstrate shock waves. These systems are of particular interest because they are experimentally accessible.