Statistics of Coherent Structures Collisions and their Dynamics on the Surface of Deep Water

        The present work is devoted to the study of coherent structures collisions dynamics in the models of deep water waves equations: the model of a supercompact equation for deep water unidirectional waves (SCEq) and the model of Dyachenko equations for potential flows of incompressible fluid with free surface. In these models there are special solutions in the form of coherent wave structures called breathers. They can be found numerically by using the Petviashvili method. One can consider the combination of such breathers as a model of rarefied soliton gas, and their paired collisions in this case are a key feature in forming of dynamics and statistics in the model. To describe statistical characteristics of breathers collision Probability Density Function (PDF) is used. PDF of breathers wave amplitudes during their collision was calculated and compared with the known results in the model of Nonlinear Schrodinger equation (NLS). In contrast to the NLS model there is a number of interesting features in the model of SCEq. For instance, the amplitude maximum of wave arising during the collision can exceed the sum of interacting breathers amplitudes, what cannot happen in NLS model. Moreover, it depends on the initial breathers steepness. In addition, it is shown that the breathers acquire phase and space shifts after each collision, and thus their velocity also changes. Depending on the relative phase breathers can give their energy or take it, and as a result their amplitude can be decreased or increased respectively. The same situation can be seen in the model of equations for potential flows of incompressible fluid with free surface. In addition to the dependence on relative phase the duration of the collision also affects the energy exchange. Breathers collisions are accompanied by appearance of little radiation, and its value is relatively less than the value of energy exchange. The results of statistics calculating and dynamics studying in the rarefied gas of coherent structures will be shown in the present work.

           The work was supported by Russian Science Foundation grant № 18-71-00079.