On the multi-component nonlinear Schrödinger equation with constant boundary conditions

The multi-component nonlinear Schrödinger equation related to C.I ' Sp(2p)/U(p) and D.III ' SO(2p)/U(p)-type symmetric spaces with non-vanishing boundary conditions is solvable with the inverse scattering method (ISM). As Lax operator L we use the generalized Zakharov-Shabat operator. We show that the ISM for the Lax operator L(x, λ) is a nonlinear analog of the Fourier-transform method. As appropriate generalizations of the usual Fourierexponential functions we use the so-called ”squared solutions”, which are constructed in terms of the fundamental analytic solutions (FAS) χ±(x, λ) of L(x, λ) and the Cartan-Weyl basis of the Lie algebra, relevant to the symmetric space. We derive the completeness relation for the ”squared solutions” which turns out to provide spectral decomposition of the recursion (generating) operators Λ±, a natural generalizations of 1 i d dx in the case of nonlinear evolution equations (NLEE).