Interactions of the vector breathers for the coupled Hirota system with 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4$$\end{document} Lax pair

In this paper, we investigate the interactions of the vector breathers for the coupled Hirota system with 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4$$\end{document} Lax pair. Firstly, we give the first-order breather solutions via the Darboux transformation, and work out the second- and third-order breather solutions with two spectral parameters via the generalized DT method. Then, by means of such solutions, we discuss the properties of the first-order vector breathers, and study the interactions of the vector breathers. We present the first-order dark-bright breather, where the one-hump bright breather can turn into a two-hump one. We find that the period of the first-order vector breather increases with the value of the real parameter in the coupled Hirota system with 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4$$\end{document} Lax pair. For two different choices of the two spectral parameters, we illustrate the interactions between two vector breathers. For one choice, in one component, we show that the one-valley dark breather interacts with a two-valley dark breather and a two-hump bright breather, respectively; In the other component, we show that two bright breathers with different amplitudes interact with each other. For the other choice, in one component, we show the interactions between two dark breathers, and between a bright breather and a dark breather, respectively; In the other component, we show that a one-hump bright breather interacts with a one-hump bright breather and a two-hump bright breather, respectively. Moreover, we exhibit the third-order vector breathers which describe the interactions among three vector breathers.