Numerical Simulation of the Soliton Turbulence in the Integrable Systems

In the theory of turbulence the statistical properties of the wave field are characterized by four moments (mean, variance, skewness, kurtosis). Important is the fourth moment, which defines the role of the large pulses, from the point of view of the wave turbulence indicates the probability of occurrence of abnormally large short-lived waves, which had a term: freak waves. Because of their evident danger to ships, oil platforms, port facilities and tourist areas on the coast, they rivet the attention of specialists from around the world. In this final qualifying work concrete examples of wave fields that contain abnormally large waves will be simulated. As an integrable system, the Gardner equation is considered, special cases of which are the Korteweg-de Vries (KdV) equation and modified Korteweg-de Vries (mKdV) equation. All numerical solutions are based on the pseudo-spectral method and were carried out by using MATLAB software tool. For the most tasks periodic boundary conditions are selected. In the theoretical section there are general information about the selected systems, as well as general historical information about the waves in nature and, in particular, the solitons. The second chapter considered of the KdV and mKdV equations, namely the features of two-soliton interaction and the bipolar soliton gas dynamics are studied. The following describes the specifics of solitons interaction in the framework of the Gardner equation, we study the wave fields of different density. Based on the results, conclusions about the possibility of abnormal pulses in the implementation are given.