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

We present a Riemann-Hilbert (RH) method directly by using the inverse scattering method (ISM) for Wadati-Konno-Ichikawa equation (WKIE) iq(t) + (q/root 1 + vertical bar q vertical bar(2))(xx) =0 The RH problem is related to two cases of scattering data: N simple poles and one Nth order pole. Under the reflection-less situation, we solve the RH problem and obtain the formulae of Nth 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. (C) 2019 Elsevier B.V. All rights reserved.