The Solutions of the NLS Equations with Self-Consistent Sources

Abstract

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.

24 pages, 3 figures, to appear in Journal of Physics A: Mathematical and General