Lax pair, Darboux-dressing transformation and localized waves of the coupled mixed derivative nonlinear Schrödinger equation in a birefringent optical fiber

Abstract

Under investigation in this paper is a coupled mixed derivative nonlinear Schrödinger equation, which can describe the pulse propagation in the femtosecond or picosecond regime of a birefringent optical fiber. Based on a Lax pair, we provide a Darboux transformation, and obtain the breather wave solutions. Then, a novel vector rogue wave solution is constructed by using Darboux-dressing transformation and asymptotic expansion. Those solutions exhibit some elastic and collision behaviors between a localized wave and a bright–dark soliton. Our works can be of importance in the theoretical experiments of the rogue wave generation mechanism and propagation trajectory.