Solitons, Breathers and Modulation Instability for a Higher-Order Coupled Nonlinear Schrödinger System for the Ultrashort Optical Pulses in a Nonlinear Medium

In this paper, we investigate some soliton and breather solutions for a higher-order coupled nonlinear Schrödinger system which may describe the ultrashort optical pulses in a nonlinear medium. With the help of the existing Darboux transformation, we construct the first- and second-order soliton solutions, as well as the first- and second-order breather solutions. We present two types of the solitons, i.e., two-hump solitons and one-hump solitons. Interaction between the two one-hump solitons, and interaction between a two-hump soliton and a one-hump soliton are presented. When the velocities of two solitons are equal, we obtain the bound state of the two solitons. Real constant ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} in the system affects the velocities of the solitons. We show the one breather and interaction between the two breathers. Velocities and shapes of the breathers are also affected by ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}. We discuss the modulation instability of that system through the linear stability analysis. That system may describe the ultrashort optical pulses in a nonlinear medium, therefore the wave phenomena proposed in our paper may provide certain theoretical references for the related experiments in the future.