N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma

Alfvén waves propagating parallel to the ambient magnetic field are modeled via the Gerdjikov-Ivanov equation. With respect to the transverse magnetic field perturbation for such an equation, we derive an N-fold generalized Darboux transformation and some semirational solutions on the constant/periodic background, where N is a positive integer. Such semirational solutions consist of the l-th-order rogue waves, the κ-l-r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \kappa -l-r\right) $$\end{document}-th-order nondegenerate breathers and the r-th-order degenerate breathers, where κ=2,3,…,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa =2,3,\ldots ,N$$\end{document}, l=0,1,2,…,κ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l=0,1,2,\ldots ,\kappa -1$$\end{document}, and r=0,2,3,…,κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=0,2,3,\ldots ,\kappa $$\end{document}. With some conditions, the l-th-order rogue wave is close to the 12(κ-l)(κ+l+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}(\kappa -l)(\kappa +l+1)$$\end{document} elements of the κ-l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \kappa -l\right) $$\end{document}-th-order breathers which locate in the center of the x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document}-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document}t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t$$\end{document} plane, and then the interaction among them forms the κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-th-order rogue wave, where x and t denote the space and time coordinates, respectively. Through the asymptotic analysis, we find that the trajectories of the second-order degenerate breathers are the logarithmic curves, and the asymptotic breathers are in good coherence with the corresponding analytic solutions in the far-field region of the x–t plane. Interaction between the second-order nondegenerate breathers and the first-order rogue wave reveals the periodic attraction and repulsion.