We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrödinger equation (AKNSr) (r = 1, 2). Using the dressing transformation (DT) method and the related tau functions we study the AKNSr 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 AKNSr system we study the properties of the focusing, defocusing and mixed focusing-defocusing versions of the so-called coupled non-linear Schrödinger 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 AKNS2 system, and three− and higher−dark−dark−soliton bound states can not exist. The AKNSr (r ≥ 3) extension is briefly discussed in this approach. The properties and calculations of some matrix elements using level one vertex operators are outlined.Dedicated to the memory of S. S. Costa.
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