Generalized AKNS System, Non-vanishing Boundary Conditions and N-Dark-Dark Solitons
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 mixe...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
13.10.2011
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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DOI: | 10.48550/arxiv.1110.3108 |