Dark-dark solitons for a set of the coupled nonlinear Schrödinger equations in a birefringent fiber

In this paper, we investigate a set of the coupled nonlinear Schrödinger equations with the four-wave mixing term, which describes the optical solitons in a birefringent fiber. Via the Kadomtsev-Petviashvili hierarchy reduction, we obtain the nondegenerate N-dark-dark soliton solutions in the Gram determinant form. Based on such solutions, we get the nondegenerate dark-dark solitons. With the analytic and graphic analysis, soliton propagating on a periodic background is presented. Via the asymptotic analysis, we find that the interaction between the two dark-dark solitons is elastic, and such an elastic interaction is shown graphically. Moreover, we also present the three dark-dark and bound-state dark-dark solitons.