Evolutionary differential inclusions as generalization of the dynamic equations describing superficial waves on water are considered. The correctness of approximation of the initial equations by means of differential inclusions is shown. Application of differential inclusions as methods of data of the initial equations to systems of the ordinary differential equations, and also applications of the received disperse dynamic systems for a substantiation of computing experiments is shown.

The methods of extreme waves modeling are discussed. The exact one-dimensional model for potential waves is used for simulation of extreme wave up to onset of breaking. The evolution of wave shape and its energy are represented. The destroying properties of wave are discussed.

The approach to generation by virtue of numerical simulations and analysis of realistic strongly nonlinear gravity wave fields is given. The waves are represented by temporal-spatial surface elevation fields. The obtained information is used with the purpose of building up spatial and temporal sequences of surface elevation, study of the spatio-temporal nonlinear wave dynamics, detailed analysis of rogue events and their evolution. The paper reports on preliminary results of the wave processing, which emphasize the rich variety of rogue wave shapes observed in the numerical simulations.

The simplest weakly nonlinear models of three-dimensional water waves are presented. The spatio-temporal spectral estimates of the New Year Wave record show a pronounced effect of the harmonic f_3/2 = 3/2 f_m, f_m − spectral peak frequency) that corresponds to the maximum of five-wave instability and a harmonic with an intermediate frequency that can be related to the well-known modulational four-wave instability. The results are discussed in the context of possible scenarios of freak wave occurrence.

Process of freak waves formation as a result of transformation of initial narrow wave spectrum considered on the base of laboratory experiments. It is demonstrated that downshifting of spectral maximum occurs during the propagation of steep initially monochromatic and bichromatic waves. This downshifting is the main reason of variability of amplitude-frequency contents of individual waves and, as sequence, the freak wave’s formation. In paper is considered how the evolution of spectrum depends on initial steepness of waves and spectral wideness.

In the article the problem of appearence of freak wave at the surface of deep water is considered. Two analytical model are proposed for two-dimensional ideal fluid. The first model is based on the conformal mapping in the ecact Euler equations of the domain occupied by the fluid to the low half-plane. In the second model canonical transformation is applied for approximate Hamiltonian. Simple nonlinear equation for normal canonical variable is derived as the result. Numerical experiments are performed to simulate freak waves formations for both models.

Interference of unidirectional swell and wind waves in deep water in frameworks of linear potential theory is considered. Wind waves are described by Pierson–Moskowitz spectrum, and swell – by the frequencymodulated wave packet. It is noticed that in case of a variable wind in a storm area the swell waves can be focused on some distance from the origin area, forming abnormal big waves («freak waves»). A visibility of the freak wave swell of different shapes in wind wave field is examined.

Formation of extreme waves (rogue waves) in a basin of constant depth is studied in the framework of nonlinear shallow water theory. It is shown that unidirectional propagation of non-breaking waves does not lead to the increase in the probability of rogue wave occurrence, though the wave field deviates from Gaussian. Wave breaking effects do not influence on this result, although in the case of large-amplitude waves the reflected wave appears and in the case of irregular wave field it may contribute to the formation of rogue wave as the result of wave collision. At the same time the collision of long irregular waves with a smooth profile and wave collision with a vertical wall increases the probability of rogue wave occurrence. The contribution of the wave breaking in this case is studied for different scenarios of wave collision for waves of different amplitudes.