  In this paper, we explore the initial-boundary value (IBV) problem for an
integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on the finite
interval by extending the Fokas unified transform approach. The solution of
this system can be expressed in terms of the solution of a 4x4 matrix
Riemann-Hilbert (RH) problem formulated in the complex k-plane. Furthermore,
the relevant jump matrices with explicit (x, t)-dependence of the matrix RH
problem can be explicitly found via three spectral functions {s(k), S(k),
S_L(k)} arising from the initial data and the Dirichlet-Neumann boundary
conditions at x=0 and x=L, respectively. The global relation is also found to
deduce two distinct but equivalent types of representations (i.e., one via the
large $k$ of asymptotics of the eigenfunctions and another one in terms of the
Gel'fand-Levitan-Marchenko (GLM) approach) for the Dirichlet and Neumann
boundary value problems. In particular, the formulae for IBV problems on the
finite interval can reduce to ones on a half-line as the length $L$ of the
interval approaches to infinity. Moreover, we also present the linearizable
boundary conditions for the GLM representations.
