Finite gap integration of the derivative nonlinear Schrödinger equation: A Riemann–Hilbert method

Abstract

In this paper we retrieve finite gap solutions of the Gerdjikov–Ivanov type derivative nonlinear Schrödinger equation by using the algebro-geometric method and the Riemann–Hilbert method. We show that the Baker–Akhiezer function of the derivative nonlinear Schrödinger equation can be described in terms of two solvable matrix Riemann–Hilbert problems on ℂ with σ2- and σ3-symmetry conditions based on the technique developed in the study of long-time asymptotics. Our main tools include matrix Baker–Akhiezer function, asymptotic analysis, algebraic curve and Riemann theta function, matrix Riemann–Hilbert problem and associated deformation procedures.