Abstract
A Wadati–Konno–Ichikawa(II)-short pulse (WKI(II)-SP) equation and its complex form are presented. Their Lax integrability are shown by constructing their corresponding Lax pairs respectively, and we proved the Liouville integrability of the real WKI(II)-SP equation. The nonlinear equation and its complex form equivalent to WKI(II)-SP equation are obtained by using hodograph transformations to carry out Darboux transformation. For the real WKI(II)-SP equation, explicit solutions are gained by Darboux transformation and generalized Darboux transformation, such as soliton solutions and higher-order semi-rational soliton solutions. For the complex WKI(II)-SP equation, we obtain breather solutions by Darboux transformation. The dynamical characteristics of the solutions are shown through several figures.
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