Rogue waves, semirational rogue waves and W-shaped solitons in the three-level coupled Maxwell–Bloch equations

Abstract

In this paper, the coupled Maxwell–Bloch equations which describe the propagation of two optical pulses in an optical medium with coherent three-level atoms are studied by Darboux transformation. Both the general nth-order rogue wave solution involving two different choices of multiple roots for the spectral characteristic equation and the multiparametric nth-order semirational solution are obtained in terms of Schur polynomials. The explicit rogue wave solutions and semirational solutions from first to second order are provided. In contrast to the bright, dark and four-petaled structures, some unusual patterns such as triple-hole, twisted-pair, composite four-petaled and composite dark rogue waves are put forward. Moreover, the interactions between dark–bright solitons and dark rogue waves and interactions between breathers and dark rogue waves are shown. Further, the higher-order nonlinear superposition modes which feature triple and quadruple temporal–spatial distributions are presented. Finally, the state transitions between rogue waves and W-shaped solitons are found where the modulation instability growth rate tends to zero under the low perturbation frequency. Particularly, the dark and double-peak W-shaped solitons are examined.