Generalized AKNS System, Non-vanishing Boundary Conditions and N-Dark-Dark Solitons

Abstract

We consider certain boundary conditions supporting soliton solutions in the generalized non-linear Schrodinger equation (AKNS). 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. 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. 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 Schrodinger 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.