  We consider the Riemann--Hilbert (RH) approach to the construction of
periodic finite-band solutions to the focusing nonlinear Schr\"odinger (NLS)
equation. An RH problem for the solution of the finite-band problem has been
recently derived via the Fokas method [1,2]. Building on this method, a
finite-band solution to the NLS equation can be given in terms of the solution
of an associated RH problem, the jump conditions for which are characterized by
specifying the endpoints of the arcs defining the contour of the RH problem and
the constants (so-called phases) involved in the jump matrices. In our work, we
solve the problem of retrieving the phases given the solution of the NLS
equation evaluated at a fixed time. Our findings are corroborated by numerical
examples of phases computation, demonstrating the viability of the method
proposed.
