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