The Darboux transformation for WKI system

Based on a conservation law, we construct a hodograph transformation for the Wadati–Konno–Ichikawa (WKI) equation, which implies that the WKI equation is equivalent to a modified WKI (mWKI) equation. Applying the Darboux transformation to the mWKI equation, we show that in both the focusing and defocusing cases, the mWKI equation admits an analytic bright soliton solution from the vacuum and the collisions of $n$ solitons are elastic based on the asymptotic analysis. In addition, we find that the mWKI equation still admits the breather and rogue wave solutions, although a modulation instability does not exist for it.