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The focusing NLS equation is the simplest universal model describing themodulation instability (MI) of quasi monochromatic waves in weakly nonlinearmedia, considered the main physical mechanism for the appearance of rogue(anomalous) waves (RWs) in Nature. In this paper we study, using the finite gapmethod, the NLS Cauchy problem for periodic initial perturbations of theunstable background solution of NLS exciting just one of the unstable modes. Wedistinguish two cases. In the case in which only the corresponding unstable gapis theoretically open, the solution describes an exact deterministic alternaterecurrence of linear and nonlinear stages of MI, and the nonlinear RW stagesare described by the 1-breather Akhmediev solution, whose parameters, differentat each RW appearance, are always given in terms of the initial data throughelementary functions. If the number of unstable modes is >1, this uniform in tdynamics is sensibly affected by perturbations due to numerics and/or realexperiments, provoking O(1) corrections to the result. In the second case inwhich more than one unstable gap is open, a detailed investigation of all thesegaps is necessary to get a uniform in $t$ dynamics, and this study is postponedto a subsequent paper. It is however possible to obtain the elementarydescription of the first nonlinear stage of MI, given again by the Akhmediev1-breather solution, and how perturbations due to numerics and/or realexperiments can affect this result.

The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. 1