  We study the time-dependent solitonic gauge fields in scalar QED, in which a
charged particle has the energy of reflectionless P\"{o}sch-Teller potential
with natural quantum numbers. Solving the quantum master equation for quadratic
correlation functions, we find the exact pair-production rates as polynomials
of inverse square of hyperbolic cosine, which exhibit solitonic characteristics
of a finite total pair production per unit volume and a non-oscillatory
behavior for the entire period, and an exponentially decaying factor in
asymptotic regions. It is shown that the solitonic gauge fields are the
simplest solutions of the quantum master equation and that the back-reaction of
the produced pairs does not destabilize the solitonic gauge fields.
