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 on $n$ and on the sign of the highest-degree coefficient $n$ in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.