ИСТИНА

Abstract. One of the basic model for rogue waves generation in the focusing Nonlinear Schrodinger equation with special Cauchy data. We show that if the initial data is a small spatially periodic perturbation of the unstable constant background, and the number of unstable modes is not too large, then the problem admits a series of simplifications: 1. Generic periodic problem is approximated by the finite-gap one, where the number of gaps is equal to the number N of the unstable modes multiplied by two. 2. For all parameters of the finite-gap solutions very elementary explicit expressions (up to $\epsilon^2$ corrections) in terms of the Cauchy data are provided. 3. The theta-series are well-approximated by finite sums of exponents (for different times different approximations are used), and at each time interval the s ${\cal N}(t)$olution is approximated by Akhmediev (${\cal N}(t) \le N$ for all t).