The Solutions of the NLS Equations with Self-Consistent Sources

We construct the generalized Darboux transformation with arbitrary functions in 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-Bäcklund 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 in time t for the Nonlinear Schrödinger 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 analyzed. PACS numbers: 02.30.lk, 05.45.yv 1

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