Bose-Einstein Condensates and Multi-Component NLS Models on Symmetric Spaces of BD.I-Type. Expansions over Squared Solutions

A special class of multicomponent NLS equations, generalizing the vector NLS and related to the BD.I-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructed thus reducing the inverse scattering problem to a RiemannHilbert problem. We introduce the minimal sets of scattering data T which determines uniquely the scattering matrix and the potential Q of the Lax operator. The elements of T can be viewed as the expansion coefficients of Q over the 'squared solutions' that are natural generalizations of the standard exponentials. Thus we demonstrate that the mapping T -> Q is a generalized Fourier transform. Special attention is paid to two special representatives of this MNLS with three-component and five components which describe spinor (F = 1 and F = 2, respectively) BoseEinstein condensates.