Riemann–Hilbert approach to the focusing and defocusing nonlocal derivative nonlinear Schrödinger equation with step-like initial data

Abstract

In this paper, we consider the Cauchy problem for the integrable nonlocal derivative nonlinear Schrödinger (DNLS) equation with step-like initial data. The main aim is to obtain a solution for the prescribed initial conditions. To do this, we adapt the Riemann–Hilbert (RH) approach to discuss of the step-like initial value problem associated with the nonlocal DNLS equation. The main idea is the representation of the solution of the Cauchy problem in terms of the solution of an associated 2 × 2 matrix RH problem and obtain the solution of the Cauchy problem of the nonlocal DNLS equation.