The higher-order positon and breather-positon solutions for the complex short pulse equation

The Darboux transformation (DT) for the coupled short pulse (cSP) equation is constructed through the lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-matrix approach, and the degenerated Darboux transformation (dDT) for the complex short pulse (CSP) equation is obtained by a conjugate reduction and degeneration limit methods. Through this dDT, we construct a series of degenerated solutions to the CSP equation: three types of higher-order positons based on vanishing boundary condition (VBC) and a smooth breather-positon (b-positon) with non-vanishing boundary condition (NVBC). Its dynamic and some new classification properties are also reviewed. Furthermore, we also studied the interaction between smooth position with three types of solitons under VBC and proved that smooth positon is a super-reflectionless potential. In addition, the generating mechanism and some characteristics of smooth b-positon were analyzed, including the spatiotemporal structure and compression effect.