Generalized AKNS System, Non-vanishing Boundary Conditions and N-Dark-Dark Solitons

• Main Authors: Assunção, A. de O, Blas, H, da Silva, M. J. B. F
• Format: Journal Article
• Language: English
• Published: 13.10.2011
• Subjects: Mathematics - Mathematical Physics; Physics - Exactly Solvable and Integrable Systems; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1110.3108

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schr\"{o}dinger equation (AKNS). Using the dressing transformation (DT) method and the related tau functions we study the AKNS$_{r}$ system for the vanishing, (constant) non-vanishing and the mixed boundary conditions, and their associated bright, dark and bright-dark N-soliton solutions, respectively. Moreover, we introduce a modified DT related to the dressing group in order to consider the free field boundary condition and derive generalized N-dark-dark solitons. We have shown that two$-$dark$-$dark$-$soliton bound states exist in the AKNS$_2$ system, and three$-$ and higher$-$dark$-$dark$-$soliton bound states can not exist. As a reduced submodel of the AKNS$_r$ system we study the properties of the focusing, defocusing and mixed focusing-defocusing versions of the so-called coupled non-linear Schr\"{o}dinger equation ($r-$CNLS), which has recently been considered in many physical applications. The properties and calculations of some matrix elements using level one vertex operators are outlined.