Darboux transformation and loop soliton solutions for the complex space–time-shifted nonlocal short pulse equation

Under investigation is the complex space–time-shifted nonlocal short pulse equation, which is connected with the complex space–time-shifted nonlocal sine-Gordon equation through a covariant reciprocal transformation. The first and the second types of Darboux transformations with respect to N different purely imaginary spectrum and 2N general complex spectrum are constructed by using loop group method, respectively. The generalized Darboux transformation corresponding to fixed number of purely imaginary spectrum with higher-order algebraic poles is proposed through limit technique. As an application, several kinds of analytical solutions including the bell-shaped loop soliton, higher-order loop soliton, breathing loop soliton and hybrid bell-shaped–breathing loop soliton solutions are obtained. It is found that the space–time-shifted parameters x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} and t0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document} can only have trivial effect of translations on the bell-shaped loop solitons, while they can produce nontrivial deformations for the breathing loop solitons. The singular solution traveling with certain space–time line is also given.