  In this paper, we propose a complex short pulse equation of both focusing and
defocusing types, which governs the propagation of ultra-short pulses in
nonlinear optical fibers. It can be viewed as an analogue of the nonlinear
Schr\"odinger (NLS) equation in the ultra-short pulse regime. Furthermore, we
construct the multi-dark soliton solution for the defocusing complex short
pulse equation through the Darboux transformation and reciprocal (hodograph)
transformation. One- and two-dark soliton solutions are given explicitly, whose
properties and dynamics are analyzed and illustrated.
