Vector rogue waves, rogue wave-to-soliton conversions and modulation instability of the higher-order matrix nonlinear Schrödinger equation

Using the 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 and Darboux-dressing transformation, we study the existence and properties of vector rogue waves of the higher-order matrix nonlinear Schrödinger equation. Our analytical results show at least three important features that are associated to rogue-wave formations and dynamics. Firstly, we show that vector rogue waves can be converted to vector solitons on mixed backgrounds. To be specific, one component of the vector soliton propagates on the constant background, while the other one propagates on the zero background. Such novel characteristics (vector solitons on mixed backgrounds) arise from higher-order effects. Secondly, the link between baseband modulation instability and rogue waves is displayed by showing the gain function. Thirdly, with the continuous-wave and mixed backgrounds, we produce a family of rational solutions for the purpose of describing the rogue waves. Bright-bright and dark-bright rogue waves with single hump and double humps are presented. We observe that the dark rogue wave could turn into the double-hump bright rogue wave. It is found that the bright rogue wave on the continuous-wave background can split up, giving birth to multiple rogue waves, while two humps of bright rogue wave on the zero background merge, giving birth to a bright rogue wave with one hump.