Revised Riemann–Hilbert problem for the derivative nonlinear Schrödinger equation: Vanishing boundary condition

With a vanishing boundary condition, we consider a revised Riemann–Hilbert problem (RHP) for the derivative nonlinear Schrödinger equation (DNLS), where an integral factor is introduced such that the RHP satisfies the normalization condition. In the reflectionless situation, we construct the formulas for the $N$th-order solutions of the DNLS equation, including the solitons and positons that respectively correspond to $N$ pairs of simple poles and one pair of $N$th-order poles of the RHP. According to the Cauchy–Binet formula, we show the expressions for $N$th-order solitons. Additionally, we give an explicit expression for the second-order positon and graphically describe evolutions of the third-order and fourth-order positons.