Riemann–Hilbert problems and soliton solutions for a multi-component cubic–quintic nonlinear Schrödinger equation

In this paper, based on the zero curvature equation, an arbitrary order matrix spectral problem is studied and its associated multi-component cubic–quintic nonlinear Schrodinger integrable hierarchy is derived. In order to solve the multi-component cubic–quintic nonlinear Schrodinger system, a class of Riemann–Hilbert problem is proposed with appropriate transformation. Through the special Riemann–Hilbert problem, where the jump matrix is considered to be an identity matrix, the soliton solutions of all integrable equations are explicitly calculated. The specific examples of one-soliton, two-soliton and N -soliton solutions are explicitly presented.