The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

• Smooth solutions of the DNLSI in determinant form with a degenerate eigenvalue.
• Smooth positon of the DNLSI, which has different property from soliton.
• Several new patterns of the higher-order rogue wave of the DNLSI.

In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schrödinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.