The Solutions of the NLS Equations with Self-Consistent Sources

Details The Solutions of the NLS Equations with Self-Consistent Sources The Solutions of the NLS Equations with Self-Consistent Sources

2004-12-29

arxiv.org/abs/nlin/0412071v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

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

Scientific paper

10.1088/0305-4470/38/11/008

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-B\"{a}cklund 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 Schr\"{o}dinger 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.

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