The mKdV equation on a finite interval

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex $k$-plane. This Riemann-Hilbert problem has explicit $(x,t)$-dependence and it involves certain functions of $k$ referred to as ``spectral functions''. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic ``global relation'' that characterize the boundary values in spectral terms.