Nonvanishing boundary conditions and dark solitons in the NLS model

We consider non-vanishing boundary conditions (NVBC) for the NLS model [6,7,27] in the context of the hybrid dressing transformation and $τ$-function approach. In order to write the NLS model in a suitable form to deal with non-vanishing boundary conditions it is introduced a new spectral parameter in such a way that the usual NLS parameter will depend on the affine parameter through the so-called Zukowsky function. In the context of the dressing transformation the introduction of the affine parameter avoids the construction of certain Riemann sheets for the usual NLS spectral parameter. In this way one introduces a Lax pair defined for the new spectral parameter and the relevant NVBC NLS $τ$ functions are obtained by the dressing transformation method. We construct the one and two dark-soliton solutions explicitly.