The N-coupled higher-order nonlinear Schrodinger equation: Riemann-Hilbert problem and multi-soliton solutions

In this work, the Riemann-Hilbert (RH) problem of the N-coupled high-order nonlinear Schrodinger (N-CHNLS) equations is studied carefully, which controls the propagation of N fields with all high-order effects such as high-order dispersion, self-steepening effect, and Raman scattering in optical fiber. The spectral analysis of the Lax pair associated with a (2N+1)x(2N+1) matrix spectral problem for the N-CHNLS equations is firstly carried out, from which a kind of RH problem is structured. Then a series of multi-soliton solutions including breather, bright, and dark solutions for the N-CHNLS equations can be formulated by the RH problem with the reflection-less case. In addition, with N=4 taken as an example, the propagation behavior of these solutions and their interactions are presented by selecting appropriate parameters with some graphics.