Hodograph transformation, various exact solutions and dynamical analysis for the complex Wadati–Konno–Ichikawa-II equation

Abstract

This paper is devoted to studying the second type of complete integrable complex Wadati–Konno–Ichikawa-II (cWKI-II) equation. We first study the conservation laws and construct the hodograph transformation for this equation. Based on the resulting transformation, we construct a new two-component associated cWKI-II equation, which is completely equivalent to the cWKI-II equation. Based on the corresponding Lax pair for this new associated equation, we construct its N-fold Darboux transformation and generalized (m,N−m)-fold Darboux transformation. Then, the associated Bäcklund transformation is used to give various exact solutions, such as smooth soliton, bursting soliton, singular-loop soliton, position controllable semi-rational soliton, breather and rogue wave solutions. The soliton structures along with dynamical evolution behaviors are discussed and shown graphically. Moreover, we investigate some important physical quantities of some soliton solutions such as amplitude, wave number, wave width, velocity, energy and phase by using the asymptotic analysis technique. Finally, we also summarize the formation reasons for some singular structures under some parameter constraints.