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We investigate the initial-boundary value problem for the integrable spin-1Gross-Pitaevskii (GP) equations with a 4x4 Lax pair on the half-line. Thesolution of this system can be obtained in terms of the solution of a 4x4matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane. Therelevant jump matrices of the RH problem can be explicitly found using the twospectral functions s(k) and S(k), which can be defined by the initial data, theDirichlet-Neumann boundary data at x=0. The global relation is establishedbetween the two dependent spectral functions. The general mappings betweenDirichlet and Neumann boundary values are analyzed in terms of the globalrelation.

An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line