On exact solutions of the Schäfer–Wayne short pulse equation: WKI eigenvalue problem

We consider a new equation recently found by Schäfer and Wayne, hereafter named Schäfer–Wayne short pulse equation (SWSPE), describing the propagation of an ultrashort pulse in nonlinear media. Using some vanishing boundary conditions, we construct and discuss the N -soliton solutions to the previous equation by means of the Wadati–Konno–Ichikawa (WKI) method, which is arguably more direct than the map through the sine-Gordon equation investigated much earlier by Sakovich and Sakovich (2005 J. Phys. Soc. Japan 74 239, 2006 J. Phys. A: Math. Gen. 39 L361 ). We particularly focus our attention on the two-soliton solution as an application. As a result, the collision process for two-soliton solutions with ‘similar’ amplitudes exhibits very different behaviour from the case when the amplitudes are ‘dissimilar’.