Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Riemann--Hilbert approach: inverse and direct problems

We consider the Riemann--Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schr\"odinger (NLS) equation, addressing the question of how the RH problem parameters can be retrieved from the solution. Within the RH approach, a finite-band solution to the NLS equation is 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.