  Boundary value problems for integrable nonlinear evolution PDEs formulated on
the finite interval can be analyzed by the unified method introduced by one of
the authors and used extensively in the literature. The implementation of this
general method to this particular class of problems yields the solution in
terms of the unique solution of a matrix Riemann-Hilbert problem formulated in
the complex $k$-plane (the Fourier plane), which has a jump matrix with
explicit $(x,t)$-dependence involving six scalar functions of $k$, called
spectral functions. Two of these functions depend on the initial data, whereas
the other four depend on all boundary values. The most difficult step of the
new method is the characterization of the latter four spectral functions in
terms of the given initial and boundary data, i.e. the elimination of the
unknown boundary values. Here, we present an effective characterization of the
spectral functions in terms of the given initial and boundary data. We present
two different characterizations of this problem. One is based on the analysis
of the so-called global relation, on the analysis of the equations obtained
from the global relation via certain transformations leaving the dispersion
relation of the associated linearized PDE invariant, and on the computation of
the large $k$ asymptotics of the eigenfunctions defining the relevant spectral
functions. The other is based on the analysis of the global relation and on the
introduction of the so-called Gelfand-Levitan-Marchenko representations of the
eigenfunctions defining the relevant spectral functions. We also show that
these two different characterizations are equivalent and that in the limit when
the length of the interval tends to infinity, the relevant formulas reduce to
the analogous formulas obtained recently for the case of boundary value
problems formulated on the half-line.

Analysis of PDEs;Exactly Solvable and Integrable Systems