Darboux transformation, bright and dark-bright solitons of an N-coupled high-order nonlinear Schrödinger system in an optical fiber

In this paper, an N -coupled high-order nonlinear Schrödinger system, which describes the properties of the ultrashort optical pulses in an optical fiber, is investigated with the help of Darboux transformation (DT) method and asymptotic analysis. Starting from the given (2 N +1)th-order Lax pair, we construct a new form of DT (complex eigenfunctions of Lax pair involved) to derive the formulas of the n th-iterated solutions, where n and N are the positive integers. On the zero background, the first- and second-order solitons are obtained and analysed through the asymptotic analysis. Multi-parameter adjustment is proceeded since there are 3 N +4 real parameters in the second-order solitons. We find that under certain conditions each of the two interaction patterns (elastic, inelastic) holds in the second-order soliton. On the plane wave background, the first-order bright and dark-bright solitons are obtained. Soliton velocities, amplitudes, widths and characteristic lines of the first-order bright and dark-bright solitons are presented and analysed.