Soliton, breather and rogue wave solutions of the coupled Gerdjikov–Ivanov equation via Darboux transformation

The coupled Gerdjikov–Ivanov (cGI) equation, associated with a 3×3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\times 3$$\end{document} matrix Lax pair, is an important integrable system that can be reduced to the third kind of derivative nonlinear Schrödinger equation, i.e., Gerdjikov–Ivanov equation. Based on the symmetric relations of the Lax pair, 2N-fold Darboux transformation for the cGI equation is constructed. As an application of the Darboux transformation, we obtain some exact solutions of the cGI equation, including bright–bright soliton, dark–bright soliton, Ma breather, breather fission, soliton fusion and dark–bright rogue wave.