The inverse scattering transform (IST) method, reformulated as a Riemann–Hilbert (RH) factorization problem, provides a powerful analytical framework for solving nonlinear integrable evolution equations. Extending this approach from the real line to finite-interval or periodic settings introduces additional challenges due to the presence of boundary data. These difficulties can be addressed by the Unified Transform Method (also known as the Fokas Method), which simultaneously analyzes both equations of the Lax pair and connects the associated spectral functions through a global relation. For certain linearizable boundary conditions, the global relation can be explicitly resolved, allowing the formulation of a Riemann–Hilbert problem in terms of given data alone.

In this talk, we present the application of this framework to the nonlinear Schrödinger equation on a finite interval with periodic boundary conditions. We show that this problem is linearizable and that its solution can be obtained in the form of a RH factorization problem whose data are determined by the scattering matrix associated with the initial data. We also briefly discuss analogous results for the Camassa–Holm equation.