Darboux transformation, localized waves and conservation laws for an M-coupled variable-coefficient nonlinear Schrödinger system in an inhomogeneous optical fiber

Optical fiber communication system is one of the supporting systems in the modern internet age. We investigate an M-coupled variable-coefficient nonlinear Schrödinger system, which describes the simultaneous pulse propagation of the M-field components in an inhomogeneous optical fiber, where M is a positive integer. With respect to the complex amplitude of the jth-field (j=1,…,M) component in the optical fiber, we construct an n-fold Darboux transformation, where n is a positive integer. Based on the n-fold Darboux transformation, we obtain some one- and two-fold localized wave solutions for the above system with the mixed defocusing-focusing-type nonlinearity and M=2. We acquire the infinitely-many conservation laws. Via such solutions, we obtain some vector gray solitons, interactions between the two vector parabolic/cubic gray solitons, and interactions between the vector parabolic/cubic breathers and gray solitons with different β(z), γ(z) and δ(z), the coefficients of the group velocity dispersion, nonlinearity and amplification/absorption. It can be found that δ(z) affects the backgrounds of the breathers and gray solitons.