  Using Lewis-Riesenfeld theory, we derive an exact non-adiabatic master
equation describing the time evolution of the QED Schwinger pair production
rate for a general time-varying electric field. This equation can be written
equivalently as a first-order matrix equation, as a Vlasov type integral
equation, or as a third-order differential equation. In the last version it
relates to the KdV equation, which allows us to construct an exact solution
using the well-known one-soliton solution to that equation. The case of
time-like delta function pulse fields is also shortly considered. We compare
with previous approaches to the purely time-dependent field case.
