Complete classification of wave structures in the Riemann problem for the third-order derivative nonlinear Schrödinger model

Abstract This paper employs Whitham modulation theory to conduct an analytical study on the Riemann problem for the third-order flow equation of a nonlinear derivative Schrödinger-type model. Through theoretical derivation, the periodic traveling wave solutions of the equation and the corresponding Whitham modulation equations are successfully obtained. By introducing self-similar variables, fundamental wave structures such as rarefaction waves and cnoidal dispersive shock waves are accurately identified. Furthermore, we provide a complete classification of all wave structures emerging from initial discontinuities, analyzing in each case the distribution of Riemann invariants and the evolution of dispersive fluid density. Verified by numerical simulations, the analytical conclusions of Whitham modulation theory are in excellent agreement with the results of direct numerical calculations.