  Solitons in one-dimensional parity-time (PT)-symmetric periodic potentials
are studied using exponential asymptotics. The new feature of this exponential
asymptotics is that, unlike conservative periodic potentials, the inner and
outer integral equations arising in this analysis are both coupled systems due
to complex-valued solitons. Solving these coupled systems, we show that two
soliton families bifurcate out from each Bloch-band edge for either
self-focusing or self-defocusing nonlinearity. An asymptotic expression for the
eigenvalues associated with the linear stability of these soliton families is
also derived. This formula shows that one of these two soliton families near
band edges is always unstable, while the other can be stable. In addition,
infinite families of PT-symmetric multi-soliton bound states are constructed by
matching the exponentially small tails from two neighboring solitons. These
analytical predictions are compared with numerics. Overall agreements are
observed, and minor differences explained.
