The system of the vortex-like structures: the viewpoint on a turbulence modelling

Abstract: In this study we suggest new approach to turbulence modeling. To develop this approach, we construct the set of the vortex-like dynamical systems evolving in the space $E_3$. These systems are constructed using the AKNS hierarchy so that they generalize thin closed vortex filament described in the local induction approximation. We construct and investigate the Hamiltonian dynamics of these systems in terms of the non - standard variables. We formulate the principles of quantization these systems focusing on the quantum description of the filaments with linearized dynamics. To describe the energy of vortex-like structures, we apply a new method based on the use of the central extended Galilei group. In a special case, we calculate the energy levels of a system of quantized vortices. The proposed approach makes it possible to describe both the interaction of such vortex - like filaments and the processes of their creation and annihilation in a natural way. As a result, we have constructed a quantum turbulent flow model that demonstrates the relationship between stochastic and integrable dynamics.