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We consider the Riemann-Hilbert (RH) approach to the construction of periodicfinite-band solutions to the focusing nonlinear Schr\"odinger (NLS) equation,addressing the question of how the RH problem parameters can be retrieved fromthe solution. Within the RH approach, a finite-band solution to the NLSequation is given in terms of the solution of an associated RH problem, thejump conditions for which are characterized by specifying the endpoints of thearcs defining the contour of the RH problem and the constants (so-calledphases) involved in the jump matrices. In our work, we solve the problem ofretrieving the phases given the solution of the NLS equation evaluated at afixed time. Our findings are corroborated by numerical examples of phasescomputation, demonstrating the viability of the method proposed.

Periodic finite-band solutions to the focusing nonlinear Schr\"{o}dinger equation by the Riemann-Hilbert approach: inverse and direct problems