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The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universalmodel describing the modulation instability (MI) of quasi monochromatic wavesin weakly nonlinear media, the main physical mechanism for the generation ofrogue (anomalous) waves (RWs) in Nature. In this paper we investigate the$x$-periodic Cauchy problem for NLS for a generic periodic initial perturbationof the unstable constant background solution, in the case of $N=1,2$ unstablemodes. We use matched asymptotic expansion techniques to show that the solutionof this problem describes an exact deterministic alternate recurrence of linearand nonlinear stages of MI, and that the nonlinear RW stages are described bythe N-breather solution of Akhmediev type, whose parameters, different at eachRW appearence, are always given in terms of the initial data through elementaryfunctions. This paper is motivated by a preceeding work of the authors in whicha different approach, the finite gap method, was used to investigate periodicCauchy problems giving rise to RW recurrence.

The exact rogue wave recurrence in the NLS periodic setting via matched asymptotic expansions, for 1 and 2 unstable modes