On Nonlocal Models of Kulish-Sklyanin Type and Generalized Fourier Transforms

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 constructing thus reducing the inverse scattering problem to a Riemann-Hilbert problem. We introduce the minimal sets of scattering data I which determines uniquely the scattering matrix and the potential Q of the Lax operator. The elements of I 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 I -> 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) Bose-Einstein condensates.