The Fokas unified transform method  (UTM) provides a novel approach for solving initial-boundary value problems (ibvp’s) for linear and integrable nonlinear partial differential equations.  In particular, it gives solution formulas for forced linear ibvp’s. Using these formulas Fokas and collaborators initiated a new approach for studying  the well-posedness in Sobolev spaces of ibvp’s for nonlinear evolution equations, which is analogous to the way well-posedness of initial value problems (ivp’s) are studied based on the Fourier method. Utilizing the Fokas solution formulas linear estimates are derived and then

using the multilinear estimates suggested by the nonlinearity it is shown that the iteration map defined by this formula is a contraction in appropriate solution spaces. In this talk we will present key points of this approach for the Korteweg-de Vries, the nonlinear Schr\”odinger and related equations and systems.

The talk is based on collaborative work with A. Fokas, D. Mantzavinos and F. Yan.