Well-posed boundary value problems for integrable evolution equations on a finite interval

We consider boundary value problems posed on an interval [0, L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x = 0 and n-N conditions at x = L, where N depends oil n and on the sign of the highest-degree coefficient alpha(n) in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solutions moreover, we show that these are the only problems that admit such a representation, These results call be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.