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

Initial-boundary value problem for integrable nonlinear evolution equations with $3\times 3$ Lax pairs on the interval