  We present an approach for analyzing initial-boundary value problems which is
formulated on the finite interval ($0\le x\le L$, where $L$ is a positive
constant) for integrable equations whose Lax pairs involve $3\times 3$
matrices. Boundary value problems for integrable nonlinear evolution PDEs can
be analyzed by the unified method introduced by Fokas and developed by him and
his collaborators. In this paper, we show that the solution can be expressed in
terms of the solution of a $3\times 3$ Riemann-Hilbert problem. The relevant
jump matrices are explicitly given in terms of the three matrix-value spectral
functions $s(k)$,$S(k)$ and $S_L(k)$, which in turn are defined in terms of the
initial values, boundary values at $x=0$ and boundary values at $x=L$,
respectively. However, these spectral functions are not independent, they
satisfy a global relation. Here, we show that the characterization of the
unknown boundary values in terms of the given initial and boundary data is
explicitly described for a nonlinear evolution PDE defined on the interval.
Also, we show that in the limit when the length of the interval tends to
infity, the relevant formulas reduce to the analogous formulas obtained for the
case of boundary value problems formulated on the half-line.
