  The unified transform method (UTM) provides a novel approach to the analysis
of initial-boundary value problems for linear as well as for a particular class
of nonlinear partial differential equations called integrable. If the latter
equations are formulated in two dimensions (either one space and one time, or
two space dimensions), the UTM expresses the solution in terms of a matrix
Riemann-Hilbert (RH) problem with explicit dependence on the independent
variables. For nonlinear integrable evolution equations, such as the celebrated
nonlinear Schr\"odinger (NLS) equation, the associated jump matrices are
computed in terms of the initial conditions and all boundary values. The
unknown boundary values are characterized in terms of the initial datum and the
given boundary conditions via the analysis of the so-called global relation. In
general, this analysis involves the solution of certain nonlinear equations. In
certain cases, called linearizable, it is possible to bypass this nonlinear
step. In these cases, the UTM solves the given initial-boundary value problem
with the same level of efficiency as the well-known inverse scattering
transform solves the initial value problem on the infinite line. We show here
that the initial-boundary value problem on a finite interval with $x$-periodic
boundary conditions (which can alternatively be viewed as the initial value
problem on a circle), belongs to the linearizable class. Indeed, by employing
certain transformations of the associated RH problem and by using the global
relation, the relevant jump matrices can be expressed explicitly in terms of
the so-called scattering data, which are computed in terms of the initial
datum. Details are given for NLS, but similar considerations are valid for
other well-known integrable evolution equations, including the Korteweg-de
Vries (KdV) and modified KdV equations.
