  We propose a new approach for the solution of initial value problems for
integrable evolution equations in the periodic setting based on the unified
transform. Using the nonlinear Schr\"odinger equation as a model example, we
show that the solution of the initial value problem on the circle can be
expressed in terms of the solution of a Riemann-Hilbert problem whose
formulation involves quantities which are defined in terms of the initial data
alone. Our approach provides an effective solution of the problem on the circle
which is conceptually analogous to the solution of the problem on the line via
the inverse scattering transform.
