The short pulse equation is integrable

We prove that the Schafer-Wayne short pulse equation (SPE), which describes the propagation of ultrashort optical pulses in nonlinear media, is integrable. First, we discover a Lax pair of the SPE that is found to be of the Wadati-Konno-Ichikawa type. Second, we construct a chain of transformations that relates the SPE with the sine-Gordon equation.