The solutions of the NLS equations with self-consistent sources

We construct the generalized Darboux transformation with arbitrary functions at time t for the AKNS equation with self-consistent sources (AKNSESCS) which, in contrast with the Darboux transformation for the AKNS equation, provides a non-auto-Backlund transformation between two AKNSESCSs with different degrees of sources. The formula for N-times repeated generalized Darboux transformation is proposed. By reduction the generalized Darboux transformation with arbitrary functions at time t for the nonlinear Schrodinger equation with self-consistent sources (NLSESCS) is obtained and enables us to find the dark soliton, bright soliton and positon solutions for NLS+ESCS and NLS−ESCS. The properties of these solution are analysed.