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 Darboux transformation (DT) method and asymptotic analysis. Starting from the given (2N + 1)th-order Lax pair, we construct a new form of the DT (with some complex eigenfunctions of a Lax pair involved) to derive the formulas of the nth-iterated solutions, where n and N are the positive integers. On the zero background, the first-and second-order solitons are obtained and analyzed through the asymptotic analysis. Multi-parameter adjustment is proceeded since there are 3N + 4 real parameters in the second-order solitons. We find that under certain conditions each of the two interaction patterns (elastic and/or inelastic) holds in the second-order solitons. On the plane wave background, the first-order bright and dark-bright solitons are obtained. Velocities, amplitudes, widths and characteristic lines of the first-order bright and dark-bright solitons are presented and analyzed.