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 {\bf 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 $\mathfrak{T}$ which determines uniquely the scattering matrix and the potential $Q$ of the Lax operator. The elements of $\mathfrak{T}$ 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 $\mathfrak{T} \to Q$ 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.