Bose-Einstein Condensates and spectral properties of multicomponent nonlinear Schrodinger equations

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We analyze the properties of the soliton solutions of a class of models describing one-dimensional BEC with spin F. We describe the minimal sets of scattering data which determine uniquely both the corresponding potential of the Lax operator and its scattering matrix. Next we give several reductions of these MNLS, derive their N-soliton solutions and analyze the soliton interactions. Finally we prove an important theorem proving that if the initial conditions satisfy the reduction then one gets a solution of the reduced MNLS.