Riemann–Hilbert approach to a new integrable nonlocal fifth-order nonlinear Schrödinger equation with step-like initial data

Abstract

In this paper, we investigate the Cauchy problem for a new integrable nonlocal fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. By solving a 2 × 2 matrix Riemann–Hilbert problem in the complex $k$-plane, we obtain the limit form solutions of the nonlocal FONLS equation. As an example, we provide an exact expression of the one-soliton solution for the nonlocal FONLS equation by the Riemann–Hilbert problem in special cases.