The N‐coupled higher‐order nonlinear Schrödinger equation: Riemann‐Hilbert problem and multi‐soliton solutions

In this work, the Riemann‐Hilbert (RH) problem of the N ‐coupled high‐order nonlinear Schrödinger ( 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 (2 N +1)×(2 N +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.