NLS bright and dark solitons in the dressing transformation and tau function approach

Abstract:
We discuss some aspects of a generalized non-linear Schrödinger equation (GNLS). This system is associated to sl(n) affine Kac-Moody algebra through a zero-curvature formulation. Using the dressing transformation method we construct the N-soliton solutions. As a reduced sub-model we obtain the so-called coupled non-linear Schrödinger equation (CNLS), which has been recently considered in many physical applications. The both vanishing and nonvanishing boundary conditions are considered, which give rise to the bright and dark solitons, respectively. The explicit computations of some matrix elements using the highest weight and the level one vertex operator representations are outlined.