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An effective integration method based on the classical solution to the Jacobiinversion problem, using Kleinian ultra-elliptic functions, is presented forquasi-periodic two-phase solutions of the focusing nonlinear Schr\"odingerequation. The two-phase solutions with real quasi-periods are known to form atwo-dimensional torus, modulo a circle of complex phase factors, that can beexpressed as a ratio of theta functions. In this paper, the two-phase solutionsare explicitly parametrized in terms of the branch points on the genus-twoRiemann surface of the theta functions. Simple formulas, in terms of theimaginary parts of the branch points, are obtained for the maximum modulus andthe minimum modulus of the two-phase solution.

Effective integration of ultra-elliptic solutions of the focusing nonlinear Schr\"odinger equation