Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2012-01-28
Nonlinear Sciences
Exactly Solvable and Integrable Systems
34 pages, 3 figures. Extended version of arXiv:1110.3108[nlin.SI], to appear in J. Phys. A: Math. Theor
Scientific paper
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.
Assun\ccão A. de O.
Blas Harold
da Silva J. B. F. M.
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