N-order bright and dark rogue waves in a Resonant erbium-doped Fibre system

The rogue waves in a resonant erbium-doped fibre system governed by a coupled system of the nonlinear Schr\"odinger equation and the Maxwell-Bloch equation (NLS-MB equations) are given explicitly by a Taylor series expansion about the breather solutions of the normalized slowly varying amplitude of the complex field envelope \(E\), polarization \(p\) and population inversion \(\eta\). The n-order breather solutions of the three fields are constructed using Darboux transformation (DT) by assuming periodic seed solutions. What is more, the n-order rogue waves are given by determinant forms with \(n+3\) free parameters. Furthermore, the possible connection between our rouge waves and the generation of supercontinuum generation is discussed.