  The focusing NLS equation is the simplest universal model describing the
modulation instability (MI) of quasi monochromatic waves in weakly nonlinear
media, considered the main physical mechanism for the appearance of rogue
(anomalous) waves (RWs) in Nature. In this paper we study, using the finite gap
method, the NLS Cauchy problem for periodic initial perturbations of the
unstable background solution of NLS exciting just one of the unstable modes. We
distinguish two cases. In the case in which only the corresponding unstable gap
is theoretically open, the solution describes an exact deterministic alternate
recurrence of linear and nonlinear stages of MI, and the nonlinear RW stages
are described by the 1-breather Akhmediev solution, whose parameters, different
at each RW appearance, are always given in terms of the initial data through
elementary functions. If the number of unstable modes is >1, this uniform in t
dynamics is sensibly affected by perturbations due to numerics and/or real
experiments, provoking O(1) corrections to the result. In the second case in
which more than one unstable gap is open, a detailed investigation of all these
gaps is necessary to get a uniform in $t$ dynamics, and this study is postponed
to a subsequent paper. It is however possible to obtain the elementary
description of the first nonlinear stage of MI, given again by the Akhmediev
1-breather solution, and how perturbations due to numerics and/or real
experiments can affect this result.
