Riemann–Hilbert method for the Wadati–Konno–Ichikawa equation: N simple poles and one higher-order pole

Abstract

Abstract We present a Riemann–Hilbert (RH) method directly by using the inverse scattering method (ISM) for Wadati–Konno–Ichikawa equation (WKIE) i q t + q 1 + | q | 2 x x = 0 . The RH problem is related to two cases of scattering data: N simple poles and one N th order pole. Under the reflection-less situation, we solve the RH problem and obtain the formulae of N th order soliton and positon solutions in the form of determinants. As applications, the first-order soliton and the second-order positon solutions are displayed in analytical and graphical ways. For the first-order soliton, it displays analytic form, bursting form, and loop form. For the second-order positon solution, an interesting inelastic phenomenon is observed that a singular positon solution including an analytic soliton and a loop soliton is split into an analytic soliton and a bursting soliton after the collision.