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The multi-component nonlinear Schrodinger equation related to C.I=Sp(2p)/U(p)and D.III=SO(2p)/U(p)-type symmetric spaces with non-vanishing boundaryconditions is solvable with the inverse scattering method (ISM). As Laxoperator L we use the generalized Zakharov-Shabat operator. We show that theISM for L is a nonlinear analog of the Fourier-transform method. As appropriategeneralizations of the usual Fourier-exponential functions we use the so-calledsquared solutions, which are constructed in terms of the fundamental analyticsolutions (FAS) of L and the Cartan-Weyl basis of the Lie algebra, relevant tothe symmetric space. We derive the completeness relation for the squaredsolutions which turns out to provide spectral decomposition of the recursion(generating) operators, a natural generalizations of id/dx in the case ofnonlinear evolution equations.

On the Multi-Component Nonlinear Schr\"odinger Equation with Constant Boundary Conditions