Temporal evolution of attractive Bose-Einstein condensate in a quasi 1D cigar-shape trap modeled through the semiclassical limit of the focusing Nonlinear Schroedinger Equation

One-dimensional (1D) Nonlinear Schroedinger Equaation (NLS) provides a good approximation to attractive Bose-Einshtein condensate (BEC) in a quasi 1D cigar-shaped optical trap in certain regimes. 1D NLS is an integrable equation that can be solved through the inverse scattering method. Our observation is that in many cases the parameters of the BEC correspond to the semiclassical (zero dispersion) limit of the focusing NLS. Hence, recent results about the strong asymptotics of the semiclassical limit solutions can be used to describe some interesting phenomena of the attractive 1D BEC. In general, the semiclassical limit of the focusing NLS exibits very strong modulation instability. However, in the case of an analytical initial data, the NLS evolution does displays some ordered structure, that can describe, for example, the bright soliton phenomenon. We discuss some general features of the semiclassical NLS evolution and propose some new observables.