Coexistence of the breather and the rogue waves for a coupled nonlinear Schrödinger equation

In this paper, based on the modified Darboux transformation, a new first-order solution of coupled fourth-order nonlinear Schrödinger equation (cNLS) is constructed. The amplitude of rogue wave, distance of the breather and the rogue wave can be changed if we adjust parameter d1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_1$$\end{document}. With the adjustment of the parameter c2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_2$$\end{document}, the breather and the rogue wave can be converted into each other, and the direction of propagation of the breather can be changed. When the initial wave height takes different values, images of the breather and the rogue wave as well as soliton-like and rogue waves can be presented.