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.