Darboux transformation and soliton solutions of the (2 +\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+$$\end{document} 1)-dimensional derivative nonlinear Schrödinger hierarchy

In this paper, we investigate the (2 +\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+$$\end{document} 1)-dimensional derivative nonlinear Schrödinger (DNLS) hierarchy. The N-fold Darboux transformation is applied to the Lax pair associated with the (2 +\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+$$\end{document} 1)-dimensional DNLS equation, and the general explicit formula for N-soliton solution is derived in terms of Vandermonde-like determinant expression. On the zero background, two different families of soliton solutions (multi-line soliton and multi-parabola soliton solutions) are constructed by applying the Darboux transformation successively. The characteristics (propagation direct, wave vector and frequency) of two types of solitons are analyzed. The interactions between line solitons, between two parabola solitons, as well as between one-line soliton with different direction and one-parabola soliton are, respectively, displayed based on the figures for several sample solutions.