New derivation of soliton solutions to the AKNS\(_2\) system via dressing transformation methods

• Published in: arXiv.org
• Main Authors: de O Assunção, A, Blas, H, M J B F da Silva
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 28.01.2012
• Subjects: Boundary conditions; Defocusing; Mathematical analysis; Operators (mathematics); Solitary waves; Transformations (mathematics)
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1201.5920

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schr\"{o}dinger equation (AKNS\(_r\))\,(\(r=1,2\)). 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. 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. 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. The AKNS\(_r\)\,(\(r\geq 3\)) extension is briefly discussed in this approach. The properties and calculations of some matrix elements using level one vertex operators are outlined.