Riemann–Hilbert approaches of an M-coupled nonlinear Schrödinger system with variable coefficients and the associated nonlocal equation

We focus on the M-coupled nonlinear Schrödinger system with variable coefficients, describing simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber. The Riemann–Hilbert problem of this system is investigated based on the (M+1)× (M+1) matrix spectral problem. The N-soliton solutions are obtained when the jump matrix is an identity matrix. A variable-coefficient nonlocal nonlinear Schrödinger equation of reverse-time type is proposed with a special reduction of the M-coupled nonlinear Schrödinger system with variable coefficients. The symmetry relations of eigenvectors for one- and two-soliton solutions are given. But it is challenging and difficult to derive such relations for N-soliton solutions when N≥ 3 . The nonlocal one- and two-solutions exhibit special dynamical properties, such as periodicity and amplitude reduction. The results in this paper might be helpful to study the related physical problem in the field of optical fiber communication.