Collisions of coherent structures such as solitons or breathers is the main mechanism of rogue waves formation in the nonlinear Schrodinger equation (NLSE) and other integrable systems. The statistical description of this process is the question of great importance for the practical applications and for the theory of integrable turbulence [1]. The full integrability of NLSE gives enormous advantages for the study of this problem. Using the direct transform one can analyse wave field in terms of spectrum for the Zakharov-Shabat system. It allows to distinguish coherent structures and to predict properties of their collisions. Meanwhile the inverse scattering transform provide a tool to generate initial conditions with given statistical properties.
In this work we present numerical simulation of modulation instability of the condensate and cnoildal wave [1,2]. For this experiments we discuss probability of rogue waves formation from randomly perturbed initial conditions in terms of breathers collisions. We propose different statistical models of solitonic and breather?s gas. One of this models is based on our recent work devoted to the theoretical and experimental study of the so called super-regular NLSE breathers [3,4]. We also discuss new criteria of rogue wave formation and different numerical algorithms for efficient calculation of N-solitonic solution. All theoretical results have obtained for the infinite line interval; we discuss their applications for description of numerical simulations with periodic boundary conditions.

The authors acknowledge support from the Russian Science Foundation (Grand No. 14-22-00174).

[1] Agafontsev, D. S., and Vladimir E. Zakharov. "Integrable turbulence and formation of rogue waves." Nonlinearity 28.8 (2015): 2791.
[2] Agafontsev, D. S., and V. E. Zakharov. "Integrable turbulence generated from modulation instability of cnoidal waves." arXiv preprint arXiv:1512.06332 (2015). Under review.
[3] Gelash, A. A., and Vladimir E. Zakharov. "Superregular solitonic solutions: a novel scenario for the nonlinear stage of modulation instability." Nonlinearity 27.4 (2014): R1.
[4] Kibler, B., Chabchoub, A., Gelash, A., Akhmediev, N., and Zakharov, V. E."Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity." Physical Review X 5.4 (2015): 041026.