  In the present paper, we propose a complex short pulse equation and a coupled
complex short equation to describe ultra-short pulse propagation in optical
fibers. They are shown to be integrable due to the existence of Lax pairs and
infinite number of conservation laws. Furthermore, we construct their
multi-soliton solutions in terms of pfaffians by virtue of Hirota bilinear
method. One- and two-soliton solutions are investigated in details, showing
favorable properties in modeling ultra-short pulses with a few optical cycles.
Especially, same as the coupled nonlinear Schrodinger equation, interactions
between two solitons are basically inelastic, which deserve further study and
has potential applications.
