Exotic Localized Vector Waves in a Two-Component Nonlinear Wave System

A new two-component nonlinear wave system is studied by the generalized perturbation (n,N-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n,N\hbox {-}n$$\end{document})-fold Darboux transformation, and various exotic localized vector waves are found. Firstly, the modulational instability is investigated to reveal the mechanism of appearance of rogue waves. Then based on the N-fold Darboux transformation, the generalized perturbation (n,N-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n,N\hbox {-}n$$\end{document})-fold Darboux transformation is constructed to solve this two-component nonlinear wave system for the first time. Finally, two types of plane-wave seed solutions are selected to explore the localized vector wave solutions such as vector periodic wave solutions, vector breather solutions, vector rogue wave solutions and vector interaction solutions. It is found that there are both localized bright-dark vector waves and bright-bright vector waves in this system, which have not been reported before.