Wednesday, February 1, 2012

1201.5920 (A. de O. Assun\ccão et al.)

New derivation of soliton solutions to the AKNS$_2$ system via dressing
transformation methods

A. de O. Assun\ccão, H. Blas, M. J. B. F. da Silva
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.
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