Dynamics of localized waves for the higher-order nonlinear Schrödinger equation with self-steepening and cubic–quintic nonlinear terms in optical fibers

Under investigation in this paper is the dynamics of localized waves for the higher-order nonlinear Schrödinger equation with self-steepening and cubic–quintic nonlinear terms, which describes the propagation of ultrashort pulses in optical fibers. Firstly, based on Lax pair, the Nth-fold Darboux transformation is constructed. Secondly, the N-soliton solutions are obtained and the interactions of solitons are analyzed graphically. Moreover, the Akhmediev breather, space-time period breather and line breather are derived. The interactions of two breathers are shown and discussed. In addition, the first-order usual rogue wave and line rogue wave are given and investigated. And the different structures of second- and third-order rogue wave are observed. Finally, the interaction solutions between rogue wave and one-breather are constructed. These results in the present work could be used to understand related physical phenomena in nonlinear optics and relevant fields.