Revised Riemann-Hilbert problem for the derivative nonlinear Schrodinger equation: Vanishing boundary condition

With a vanishing boundary condition, we consider a revised Riemann-Hilbert problem (RHP) for the derivative nonlinear Schr & ouml;dinger 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.