ИСТИНА

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. The generic periodic problem is approximated by a 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 in terms of the Cauchy data are provided, to the leading order. 3. The theta-series are well-approximated by finite sums of exponents(for different times different approximations are used), and in each time interval the solution is well-described by the n(t)-soliton solution (n<= N for all t) of Akhmediev type.